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Computers & Mathematics with Applications
Journal Prestige (SJR): 1.058 Citation Impact (citeScore): 2 Number of Followers: 11 Subscription journal ISSN (Print) 08981221 Published by Elsevier [3161 journals] 
 Superconvergence analysis of a twogrid method for nonlinear hyperbolic
equations Abstract: Publication date: Available online 14 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Yifan Wei, Dongyang ShiAbstractIn this paper, the superconvergence analysis of a secondorder fully discrete scheme with twogrid method (TGM) is studied for the twodimensional nonlinear hyperbolic equations by the bilinear finite element. The existence and uniqueness of the solution for the scheme are proved rigorously. By use of the combination technique of the interpolation and Ritz projection, the superclose results in H1norm are obtained, and the global superconvergence is derived through the interpolated postprocessing approach. Finally, some numerical results are provided to verify the theoretical predictions and show the efficiency of the proposed TGM.
 Abstract: Publication date: Available online 14 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Yifan Wei, Dongyang ShiAbstractIn this paper, the superconvergence analysis of a secondorder fully discrete scheme with twogrid method (TGM) is studied for the twodimensional nonlinear hyperbolic equations by the bilinear finite element. The existence and uniqueness of the solution for the scheme are proved rigorously. By use of the combination technique of the interpolation and Ritz projection, the superclose results in H1norm are obtained, and the global superconvergence is derived through the interpolated postprocessing approach. Finally, some numerical results are provided to verify the theoretical predictions and show the efficiency of the proposed TGM.
 On quasi shiftsplitting iteration method for a class of saddle point
problems Abstract: Publication date: Available online 14 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): WenLi Gao, XiAn Li, XinMing LuAbstractMotivated by a technique of scalesplitting for matrix, we put forward a quasi shiftsplitting (named after QSS) iteration method to address a class of large scale sparse saddle point problems, this novel method also naturally leads to the corresponding QSSpreconditioner. In addition, an extrapolated variant of QSS (named after EQSS) iteration method is constructed. Further, some useful convergence properties of the QSS and EQSS iteration methods and spectral properties of QSSpreconditioner are studied. Finally, a saddle point equations stem from model Navier–Stokes problem is supplied to illustrate the effectiveness of our new two methods and QSSpreconditioner.
 Abstract: Publication date: Available online 14 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): WenLi Gao, XiAn Li, XinMing LuAbstractMotivated by a technique of scalesplitting for matrix, we put forward a quasi shiftsplitting (named after QSS) iteration method to address a class of large scale sparse saddle point problems, this novel method also naturally leads to the corresponding QSSpreconditioner. In addition, an extrapolated variant of QSS (named after EQSS) iteration method is constructed. Further, some useful convergence properties of the QSS and EQSS iteration methods and spectral properties of QSSpreconditioner are studied. Finally, a saddle point equations stem from model Navier–Stokes problem is supplied to illustrate the effectiveness of our new two methods and QSSpreconditioner.
 Numerical study on flow field and pollutant dispersion in an ideal street
canyon within a real tree model at different wind velocities Abstract: Publication date: Available online 10 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Le Wang, Junwei Su, Zhaolin Gu, Liyu TangAbstractStudying the effects of trees on airflow and pollutant dispersion in urban street canyons is of considerable significance to clarify the laws of urban microscale airflow and pollutant dispersion. To characterize the trees in street canyons, different from the traditional vegetation resistance source method, in this paper, a fine tree model is used for numerical simulations of the airflow and pollutant dispersion in street canyons at varying inflow wind velocities. It is found during the study that the presence of trees physically blocks the airflow in street canyons. The airflow is sheared by the trunk, canopy, or branches, and then circumvents them, especially at a high inflow wind velocity. The lowvelocity area in the street canyon distributes to the leeward side of the tree trunk as well as in the canopy area, and a discrete lowvelocity distribution exists mainly in the canopy area. The average wind velocity in a street canyon with trees is approximately 39.5% lower at an inflow wind velocity of 1.7 m/s than that in a canyon without trees. In the presence of trees, the pollutant concentration in street canyons increases significantly, and the pollutants significantly accumulate between the tree trunk and the leeward side. With increasing inflow velocity, the pollutant concentration in a street canyon constantly changes but is much higher than that in the absence of trees. At a wind velocity of 5.7 m/s, the average pollutant concentration is 18.6% higher in street canyons with trees than in those without trees.
 Abstract: Publication date: Available online 10 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Le Wang, Junwei Su, Zhaolin Gu, Liyu TangAbstractStudying the effects of trees on airflow and pollutant dispersion in urban street canyons is of considerable significance to clarify the laws of urban microscale airflow and pollutant dispersion. To characterize the trees in street canyons, different from the traditional vegetation resistance source method, in this paper, a fine tree model is used for numerical simulations of the airflow and pollutant dispersion in street canyons at varying inflow wind velocities. It is found during the study that the presence of trees physically blocks the airflow in street canyons. The airflow is sheared by the trunk, canopy, or branches, and then circumvents them, especially at a high inflow wind velocity. The lowvelocity area in the street canyon distributes to the leeward side of the tree trunk as well as in the canopy area, and a discrete lowvelocity distribution exists mainly in the canopy area. The average wind velocity in a street canyon with trees is approximately 39.5% lower at an inflow wind velocity of 1.7 m/s than that in a canyon without trees. In the presence of trees, the pollutant concentration in street canyons increases significantly, and the pollutants significantly accumulate between the tree trunk and the leeward side. With increasing inflow velocity, the pollutant concentration in a street canyon constantly changes but is much higher than that in the absence of trees. At a wind velocity of 5.7 m/s, the average pollutant concentration is 18.6% higher in street canyons with trees than in those without trees.
 ERHSS iteration method for PDE optimal control problem
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): ZhenWei Sun, Li WangAbstractAn extended regularized Hermitian and skewHermitian splitting iteration method, for solving the double saddle point linear system arising from PDE optimal control problem, is proposed. The unconditional convergence of the method is proved. Furthermore, some spectral properties of the preconditioned matrix are investigated. Numerical experiments show that the iteration method is feasible and efficient.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): ZhenWei Sun, Li WangAbstractAn extended regularized Hermitian and skewHermitian splitting iteration method, for solving the double saddle point linear system arising from PDE optimal control problem, is proposed. The unconditional convergence of the method is proved. Furthermore, some spectral properties of the preconditioned matrix are investigated. Numerical experiments show that the iteration method is feasible and efficient.
 Direct meshless local Petrov–Galerkin (DMLPG) method for timefractional
fourthorder reaction–diffusion problem on complex domains Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Mostafa Abbaszadeh, Mehdi DehghanAbstractA new numerical scheme has been developed based on the fast and efficient meshless local weak form i.e direct meshless local Petrov–Galerkin (DMLPG) method for solving the fractional fourthorder partial differential equation on computational domains with complex shape. The fractional derivative is the Riemann–Liouville fractional derivative. At first, a finite difference scheme with the secondorder accuracy has been employed to discrete the time variable. Then, the DMLPG technique is employed to achieve a fulldiscrete scheme. The timediscrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the L2 space. Also, some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and nonregular grid points.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Mostafa Abbaszadeh, Mehdi DehghanAbstractA new numerical scheme has been developed based on the fast and efficient meshless local weak form i.e direct meshless local Petrov–Galerkin (DMLPG) method for solving the fractional fourthorder partial differential equation on computational domains with complex shape. The fractional derivative is the Riemann–Liouville fractional derivative. At first, a finite difference scheme with the secondorder accuracy has been employed to discrete the time variable. Then, the DMLPG technique is employed to achieve a fulldiscrete scheme. The timediscrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the L2 space. Also, some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and nonregular grid points.
 Numerical analysis of two grad–div stabilization methods for the
timedependent Stokes/Darcy model Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Yi Qin, Yanren Hou, Pengzhan Huang, Yongshuai WangAbstractWe aim to present two algorithms for the nonstationary Stokes/Darcy model. The first one is the standard grad–div stabilization scheme. The other one is a modular grad–div based on the standard Backward Euler code which does not crash or slow down for large value grad–div parameters. Both algorithms cannot only improve the efficiency and accuracy of calculation but also can improve mass conservation, while the modular algorithm can be better. We give a complete theoretical analysis of the stability and error estimations of the algorithms. Finally, the theoretical results are verified by numerical experiments and the advantages of adding grad–div stabilization terms are demonstrated.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Yi Qin, Yanren Hou, Pengzhan Huang, Yongshuai WangAbstractWe aim to present two algorithms for the nonstationary Stokes/Darcy model. The first one is the standard grad–div stabilization scheme. The other one is a modular grad–div based on the standard Backward Euler code which does not crash or slow down for large value grad–div parameters. Both algorithms cannot only improve the efficiency and accuracy of calculation but also can improve mass conservation, while the modular algorithm can be better. We give a complete theoretical analysis of the stability and error estimations of the algorithms. Finally, the theoretical results are verified by numerical experiments and the advantages of adding grad–div stabilization terms are demonstrated.
 Meshless RBFs method for numerical solutions of twodimensional high order
fractional Sobolev equations Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Manzoor Hussain, Sirajul Haq, Abdul GhafoorAbstractIn this paper, meshless RBFs method is proposed to solve twodimensional timefractional Sobolev equations. The proposed method uses RBFs for approximation of spatial operator. Finite difference formula of O(δt2−α)(0
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Manzoor Hussain, Sirajul Haq, Abdul GhafoorAbstractIn this paper, meshless RBFs method is proposed to solve twodimensional timefractional Sobolev equations. The proposed method uses RBFs for approximation of spatial operator. Finite difference formula of O(δt2−α)(0
 Adaptive finite element method for the sound wave problems in two kinds of
media Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Hao Wang, Wei Yang, Yunqing HuangAbstractIn this paper, we consider the adaptive finite element method for sound wave propagation problems in two kinds of media, which are the linear anisotropic acoustic materials (the cloak metamaterials and penetrable media) and the nonlinear acoustic materials. A posteriori error estimator based on the new flux recovery technique and a residual type posteriori error estimator are proposed. Based on our a posteriori error estimators, the adaptive finite element algorithm is given for numerical simulations of the Helmholtz equations in different media. Extensive numerical results demonstrate the effectiveness of the adaptive algorithm.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Hao Wang, Wei Yang, Yunqing HuangAbstractIn this paper, we consider the adaptive finite element method for sound wave propagation problems in two kinds of media, which are the linear anisotropic acoustic materials (the cloak metamaterials and penetrable media) and the nonlinear acoustic materials. A posteriori error estimator based on the new flux recovery technique and a residual type posteriori error estimator are proposed. Based on our a posteriori error estimators, the adaptive finite element algorithm is given for numerical simulations of the Helmholtz equations in different media. Extensive numerical results demonstrate the effectiveness of the adaptive algorithm.
 Linear second order energy stable schemes for phase field crystal growth
models with nonlocal constraints Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Xiaobo Jing, Qi WangAbstractWe present a set of linear, second order, unconditionally energy stable schemes for the Allen–Cahn model with nonlocal constraints for crystal growth that conserves the mass of each phase. Solvability conditions are established for the linear systems resulting from the schemes. Convergence rates are verified numerically. Dynamics obtained using the Allen–Cahn model with nonlocal constraints are compared with the one obtained using the classic Allen–Cahn model as well as the Cahn–Hilliard model, respectively, demonstrating slower dynamics than that of the Allen–Cahn model but faster dynamics than that of the Cahn–Hilliard model. Thus, the Allen–Cahn model with nonlocal constraints can serve as an alternative to the Cahn–Hilliard model in simulating crystal growth while conserving the mass of each phase. Two Benchmark examples are presented to contrast the predictions made with the four models, highlighting the accuracy and effectiveness of the Allen–Cahn model with nonlocal constraints.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Xiaobo Jing, Qi WangAbstractWe present a set of linear, second order, unconditionally energy stable schemes for the Allen–Cahn model with nonlocal constraints for crystal growth that conserves the mass of each phase. Solvability conditions are established for the linear systems resulting from the schemes. Convergence rates are verified numerically. Dynamics obtained using the Allen–Cahn model with nonlocal constraints are compared with the one obtained using the classic Allen–Cahn model as well as the Cahn–Hilliard model, respectively, demonstrating slower dynamics than that of the Allen–Cahn model but faster dynamics than that of the Cahn–Hilliard model. Thus, the Allen–Cahn model with nonlocal constraints can serve as an alternative to the Cahn–Hilliard model in simulating crystal growth while conserving the mass of each phase. Two Benchmark examples are presented to contrast the predictions made with the four models, highlighting the accuracy and effectiveness of the Allen–Cahn model with nonlocal constraints.
 Primal–dual weak Galerkin finite element methods for elliptic Cauchy
problems Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Chunmei Wang, Junping WangAbstractThe authors propose and analyze a wellposed numerical scheme for a type of illposed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, wellposed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak L2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Chunmei Wang, Junping WangAbstractThe authors propose and analyze a wellposed numerical scheme for a type of illposed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler–Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal–dual weak Galerkin finite element method. This new primal–dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, wellposed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak L2 topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.
 Asymptotic stability of a stochastic May mutualism system
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Guodong Liu, Haokun Qi, Zhengbo Chang, Xinzhu MengAbstractThis paper deals with a mutualism system in random environments, in which the cooperation of two species is guaranteed depending on increasing the carrying capacity of each other. Mathematically, we obtain the existence and uniqueness of a stable stationary distribution by means of Markov semigroup theory and Fokker–Planck equation. In addition, we prove that densities of the distributions for the solutions converge in L1 to an invariant density under appropriate conditions. Numerical simulations are also presented to illustrate the theoretical results.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Guodong Liu, Haokun Qi, Zhengbo Chang, Xinzhu MengAbstractThis paper deals with a mutualism system in random environments, in which the cooperation of two species is guaranteed depending on increasing the carrying capacity of each other. Mathematically, we obtain the existence and uniqueness of a stable stationary distribution by means of Markov semigroup theory and Fokker–Planck equation. In addition, we prove that densities of the distributions for the solutions converge in L1 to an invariant density under appropriate conditions. Numerical simulations are also presented to illustrate the theoretical results.
 Maximizing expected terminal utility of an insurer with high gain tax by
investment and reinsurance Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Lin Xu, Shaosheng Xu, Dingjun YaoAbstractThis paper investigates optimal investment and proportional reinsurance policies for an insurer who subjects to pay high gain tax. The surplus process of the insurer and the return process of the financial market are both modulated by the external macroeconomic environment. The dynamic of the external macroeconomic environment is specified by a Markov chain with finite states. Once the insurer’s accumulated profits attain a new maximum, they have to pay high gain tax. The objective of the insurer is to maximize the expected terminal utility by investment and reinsurance. The controlled wealth process of the insurer turned out to be a controlled jump diffusion process with reflections and Markov regime switching. By the weak dynamic programming principle (WDPP), we prove that the value function is the unique viscosity solution to the coupled HamiltonJacobBellman (HJB) equations with first derivative boundary constraints. By the Markov chain approximating method for the HJB equations, we construct a numerical scheme for approximating the viscosity solution to the coupled HJB equations. Two numerical examples are presented to illustrate the impact of both high gain tax and regime switching on the optimal policies.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Lin Xu, Shaosheng Xu, Dingjun YaoAbstractThis paper investigates optimal investment and proportional reinsurance policies for an insurer who subjects to pay high gain tax. The surplus process of the insurer and the return process of the financial market are both modulated by the external macroeconomic environment. The dynamic of the external macroeconomic environment is specified by a Markov chain with finite states. Once the insurer’s accumulated profits attain a new maximum, they have to pay high gain tax. The objective of the insurer is to maximize the expected terminal utility by investment and reinsurance. The controlled wealth process of the insurer turned out to be a controlled jump diffusion process with reflections and Markov regime switching. By the weak dynamic programming principle (WDPP), we prove that the value function is the unique viscosity solution to the coupled HamiltonJacobBellman (HJB) equations with first derivative boundary constraints. By the Markov chain approximating method for the HJB equations, we construct a numerical scheme for approximating the viscosity solution to the coupled HJB equations. Two numerical examples are presented to illustrate the impact of both high gain tax and regime switching on the optimal policies.
 Interval tensors and their application in solving multilinear systems of
equations Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Hassan Bozorgmanesh, Masoud Hajarian, Anthony Theodore ChronopoulosAbstractIn this paper, we introduce interval tensors and present some results about their eigenvalues, positive definiteness and application in solving multilinear systems. It is proved that the set of maximum Zeigenvalues of a symmetric interval tensor is a compact interval. Also, several bounds for eigenvalues of an interval tensor are proposed. In addition, necessary and sufficient conditions for having a positive definite interval tensor are presented and investigated. Furthermore, solving tensor equations using interval methods is presented and the interval Jacobi and Gauss–Seidel algorithms are extended for interval multilinear systems. Finally, some numerical experiments are carried out to illustrate the methods.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Hassan Bozorgmanesh, Masoud Hajarian, Anthony Theodore ChronopoulosAbstractIn this paper, we introduce interval tensors and present some results about their eigenvalues, positive definiteness and application in solving multilinear systems. It is proved that the set of maximum Zeigenvalues of a symmetric interval tensor is a compact interval. Also, several bounds for eigenvalues of an interval tensor are proposed. In addition, necessary and sufficient conditions for having a positive definite interval tensor are presented and investigated. Furthermore, solving tensor equations using interval methods is presented and the interval Jacobi and Gauss–Seidel algorithms are extended for interval multilinear systems. Finally, some numerical experiments are carried out to illustrate the methods.
 A highorder finite volume method with improved isotherms reconstruction
for the computation of multiphase flows using the
Navier–Stokes–Korteweg equations Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Abel Martínez, Luis Ramírez, Xesús Nogueira, Sofiane Khelladi, Fermín NavarrinaAbstractIn this work we solve the Navier–Stokes–Korteweg (NSK) equations to simulate a twophase fluid with phase change. We use these equations on a diffuse interface approach, where the properties of the fluid vary continuously across the interface that separates the different phases. The model is able to describe the behavior of both phases with the same set of equations, and it is also able to handle problems with great changes in the topology of the problem. However, highorder derivatives are present in NSK equations, which is a difficulty for the design of a numerical method to solve the problem. Here, we propose the use of a highorder Finite Volume method with Moving Least Squares approximations to handle highorder derivatives and solve the NSK equations. Moreover, a new methodology to obtain accurate equations of state is presented. In this method, we use any accurate equation of state for the pure phases. Under the saturation curve, a Bspline reconstruction fulfilling a given set of thermodynamic criteria is performed. The new EOS can be used for computations using diffuse interface modeling. Several numerical examples to show the accuracy of the new approach are presented.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Abel Martínez, Luis Ramírez, Xesús Nogueira, Sofiane Khelladi, Fermín NavarrinaAbstractIn this work we solve the Navier–Stokes–Korteweg (NSK) equations to simulate a twophase fluid with phase change. We use these equations on a diffuse interface approach, where the properties of the fluid vary continuously across the interface that separates the different phases. The model is able to describe the behavior of both phases with the same set of equations, and it is also able to handle problems with great changes in the topology of the problem. However, highorder derivatives are present in NSK equations, which is a difficulty for the design of a numerical method to solve the problem. Here, we propose the use of a highorder Finite Volume method with Moving Least Squares approximations to handle highorder derivatives and solve the NSK equations. Moreover, a new methodology to obtain accurate equations of state is presented. In this method, we use any accurate equation of state for the pure phases. Under the saturation curve, a Bspline reconstruction fulfilling a given set of thermodynamic criteria is performed. The new EOS can be used for computations using diffuse interface modeling. Several numerical examples to show the accuracy of the new approach are presented.
 Projectionbased reduced order models for a cut finite element method in
parametrized domains Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Efthymios N. Karatzas, Francesco Ballarin, Gianluigi RozzaAbstractThis work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projectionbased reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Efthymios N. Karatzas, Francesco Ballarin, Gianluigi RozzaAbstractThis work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projectionbased reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail.
 An economic crossdiffusion mutualistic model for cities emergence
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Gonzalo F. deCórdoba, Gonzalo GalianoAbstractWe study an evolution crossdiffusion problem with mutualistic Lotka–Volterra reaction term to modelize the longterm spatial distribution of labor and capital. The mutualistic behavior is deduced from the gradient flow associated to profits maximization. We perform a linear and weakly nonlinear stability analysis and find conditions under which the uniform profits optimum becomes unstable, leading to pattern formation. The patterns alternate regions of high and low concentrations of labor and capital, which may be interpreted as cities. Finally, numerical simulations based on the weakly nonlinear analysis, as well as in a finite element approximation, are provided.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Gonzalo F. deCórdoba, Gonzalo GalianoAbstractWe study an evolution crossdiffusion problem with mutualistic Lotka–Volterra reaction term to modelize the longterm spatial distribution of labor and capital. The mutualistic behavior is deduced from the gradient flow associated to profits maximization. We perform a linear and weakly nonlinear stability analysis and find conditions under which the uniform profits optimum becomes unstable, leading to pattern formation. The patterns alternate regions of high and low concentrations of labor and capital, which may be interpreted as cities. Finally, numerical simulations based on the weakly nonlinear analysis, as well as in a finite element approximation, are provided.

( h − h ∕ 2 )
( h − h ∕ 2 ) type error estimators Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Christoph Erath, Gregor Gantner, Dirk PraetoriusAbstractFor some Poissontype model problem, we prove that adaptive FEM driven by the (h−h∕2)type error estimators from FerrazLeite et al. (2010) leads to convergence with optimal algebraic convergence rates. Besides the implementational simplicity, another striking feature of these estimators is that they can provide guaranteed lower bounds for the energy error with known efficiency constant 1.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Christoph Erath, Gregor Gantner, Dirk PraetoriusAbstractFor some Poissontype model problem, we prove that adaptive FEM driven by the (h−h∕2)type error estimators from FerrazLeite et al. (2010) leads to convergence with optimal algebraic convergence rates. Besides the implementational simplicity, another striking feature of these estimators is that they can provide guaranteed lower bounds for the energy error with known efficiency constant 1.
 Simulation of particles dissolution in the shear flow: A combined
concentration lattice Boltzmann and smoothed profile approach Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Raziyeh Safa, Ataallah Soltani Goharrizi, Saeed Jafari, Ebrahim Jahanshahi JavaranAbstractIn the present study, the combination of the Concentration Lattice Boltzmann Method and the Smoothed Profile Method (CLBM–SPM) was used to simulate the dissolution of circular particles between parallel plates moving in opposite directions. The hydrodynamics and fluid concentration simulation were performed based on the single relaxation time Lattice Boltzmann Method. LBM convection–diffusion equation was then used to solve the concentration of the solute in the fluid phase. Additionally, SPM was employed to apply the noslip boundary condition at the solid–fluid interface and to calculate the concentration forces. Initially, the results of the numerical solution were compared with the ones presented in the literature. Then the effects of the initial solid volume fraction, the Schmidt number, the Reynolds number, and particle size were studied to examine the behavior of particles dissolution. The results showed that the smallest dissolution time in the systems with different volume fractions was in a one with the least solid volume fraction. As the volume fraction was increased, the solid–fluid mass transfer driving force was decreased in the system. Also, with the rise of the Schmidt number, the dissolution time was increased, due to the decrease of the diffusion coefficient of the fluid flow. Moreover, by increasing the Reynolds number, the time required for the volume fraction ratio to reach 0.05 of its initial value was reduced. Finally, the particle size in this system was studied. The results indicated that with the decrease in particle size (or increase in the surface area), we could significantly alter the dissolution time.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Raziyeh Safa, Ataallah Soltani Goharrizi, Saeed Jafari, Ebrahim Jahanshahi JavaranAbstractIn the present study, the combination of the Concentration Lattice Boltzmann Method and the Smoothed Profile Method (CLBM–SPM) was used to simulate the dissolution of circular particles between parallel plates moving in opposite directions. The hydrodynamics and fluid concentration simulation were performed based on the single relaxation time Lattice Boltzmann Method. LBM convection–diffusion equation was then used to solve the concentration of the solute in the fluid phase. Additionally, SPM was employed to apply the noslip boundary condition at the solid–fluid interface and to calculate the concentration forces. Initially, the results of the numerical solution were compared with the ones presented in the literature. Then the effects of the initial solid volume fraction, the Schmidt number, the Reynolds number, and particle size were studied to examine the behavior of particles dissolution. The results showed that the smallest dissolution time in the systems with different volume fractions was in a one with the least solid volume fraction. As the volume fraction was increased, the solid–fluid mass transfer driving force was decreased in the system. Also, with the rise of the Schmidt number, the dissolution time was increased, due to the decrease of the diffusion coefficient of the fluid flow. Moreover, by increasing the Reynolds number, the time required for the volume fraction ratio to reach 0.05 of its initial value was reduced. Finally, the particle size in this system was studied. The results indicated that with the decrease in particle size (or increase in the surface area), we could significantly alter the dissolution time.
 Analysis of a continuous Galerkin method with mesh modification for
twodimensional telegraph equation Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Zhihui Zhao, Hong Li, Yang LiuAbstractIn this paper, we investigate the convergence of a space–time continuous Galerkin (STCG) method for twodimensional (2D) telegraph equation. The variable time steps and spatial mesh structures are allowed, which are necessary for adaptive computations on unstructured mesh. We demonstrate existence and uniqueness of the numerical solution and give a priori estimate in L∞(L2) norm without space–time mesh restrictions. The analysis method presented here develops and improves the existing theories so that it can be used to study the more general timedependent partial differential equations (PDEs). Some numerical experiments are provided, and compared with the standard finite element (FE) method to confirm the high efficiency and the high accuracy for the proposed scheme.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Zhihui Zhao, Hong Li, Yang LiuAbstractIn this paper, we investigate the convergence of a space–time continuous Galerkin (STCG) method for twodimensional (2D) telegraph equation. The variable time steps and spatial mesh structures are allowed, which are necessary for adaptive computations on unstructured mesh. We demonstrate existence and uniqueness of the numerical solution and give a priori estimate in L∞(L2) norm without space–time mesh restrictions. The analysis method presented here develops and improves the existing theories so that it can be used to study the more general timedependent partial differential equations (PDEs). Some numerical experiments are provided, and compared with the standard finite element (FE) method to confirm the high efficiency and the high accuracy for the proposed scheme.
 Lump, mixed lumpstripe and rogue wavestripe solutions of a
(3+1)dimensional nonlinear wave equation for a liquid with gas bubbles Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Meng Wang, Bo Tian, Yan Sun, Ze ZhangAbstractLiquids with gas bubbles are very common in science, engineering, physics, nature and life. Under investigation in this paper is a (3+1)dimensional nonlinear wave equation for a liquid with gas bubbles. With respect to the velocity of the liquidgas bubble mixture, we obtain the lump, rogue wave, mixed lumpstripe soliton and mixed rogue wavestripe soliton solutions via the symbolic computation. Based on the mixed lumpstripe soliton solutions, we construct the rogue wave and investigate the fusion and fission phenomena between a lump and the onestripe soliton. We graphically study the mixed lumpstripe soliton under the influence of the parameters β, γ, δ and ξ, which represent the dispersion, perturbed effect, disturbed wave velocities along the y and z (i.e., the two transverse) directions, respectively. With the decreasing value of β to 1, the graph from a lump and onestripe soliton shows a soliton; with the increasing value of γ to 3, location of the lump moves along the negative x axis; with the value of δ increasing to 0.5, location of the lump moves along the positive x axis; with the increasing value of ξ to 3, location and range of the lump soliton keep unchanged. With respect to the velocity of the mixture, we obtain the interaction between a rogue wave and a pair of stripe solitons according to mixed rogue wavestripe soliton solutions. A lump is provided.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): Meng Wang, Bo Tian, Yan Sun, Ze ZhangAbstractLiquids with gas bubbles are very common in science, engineering, physics, nature and life. Under investigation in this paper is a (3+1)dimensional nonlinear wave equation for a liquid with gas bubbles. With respect to the velocity of the liquidgas bubble mixture, we obtain the lump, rogue wave, mixed lumpstripe soliton and mixed rogue wavestripe soliton solutions via the symbolic computation. Based on the mixed lumpstripe soliton solutions, we construct the rogue wave and investigate the fusion and fission phenomena between a lump and the onestripe soliton. We graphically study the mixed lumpstripe soliton under the influence of the parameters β, γ, δ and ξ, which represent the dispersion, perturbed effect, disturbed wave velocities along the y and z (i.e., the two transverse) directions, respectively. With the decreasing value of β to 1, the graph from a lump and onestripe soliton shows a soliton; with the increasing value of γ to 3, location of the lump moves along the negative x axis; with the value of δ increasing to 0.5, location of the lump moves along the positive x axis; with the increasing value of ξ to 3, location and range of the lump soliton keep unchanged. With respect to the velocity of the mixture, we obtain the interaction between a rogue wave and a pair of stripe solitons according to mixed rogue wavestripe soliton solutions. A lump is provided.
 On anti bounce back boundary condition for lattice Boltzmann schemes
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): François Dubois, Pierre Lallemand, Mohamed Mahdi TekitekAbstractIn this contribution, we recall the derivation of the anti bounce back boundary condition for the D2Q9 lattice Boltzmann scheme. We recall various elements of the state of the art for anti bounce back applied to linear heat and acoustics equations and in particular the possibility to take into account curved boundaries. We present an asymptotic analysis that allows an expansion of all the fields in the boundary cells. This analysis based on pure Taylor expansions confirms the well known behaviour of anti bounce back boundary for the heat equation. The analysis puts also in evidence a hidden differential boundary condition in the case of linear acoustics. Indeed, we observe discrepancies in the first layers near the boundary. To reduce these discrepancies, we propose a new boundary condition mixing bounce back for the oblique links and anti bounce back for the normal link. This boundary condition is able to enforce both pressure and tangential velocity on the boundary. Numerical tests for the Poiseuille flow illustrate our theoretical analysis and show improvements in the quality of the flow.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): François Dubois, Pierre Lallemand, Mohamed Mahdi TekitekAbstractIn this contribution, we recall the derivation of the anti bounce back boundary condition for the D2Q9 lattice Boltzmann scheme. We recall various elements of the state of the art for anti bounce back applied to linear heat and acoustics equations and in particular the possibility to take into account curved boundaries. We present an asymptotic analysis that allows an expansion of all the fields in the boundary cells. This analysis based on pure Taylor expansions confirms the well known behaviour of anti bounce back boundary for the heat equation. The analysis puts also in evidence a hidden differential boundary condition in the case of linear acoustics. Indeed, we observe discrepancies in the first layers near the boundary. To reduce these discrepancies, we propose a new boundary condition mixing bounce back for the oblique links and anti bounce back for the normal link. This boundary condition is able to enforce both pressure and tangential velocity on the boundary. Numerical tests for the Poiseuille flow illustrate our theoretical analysis and show improvements in the quality of the flow.
 Modelling the effect of particle inertia on the orientation kinematics of
fibres and spheroids immersed in a simple shear flow Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): A. Scheuer, G. Grégoire, E. AbissetChavanne, F. Chinesta, R. KeuningsAbstractSimulations of flows containing nonspherical particles (fibres or ellipsoids) rely on the knowledge of the equation governing the particle motion in the flow. Most models used nowadays are based on the pioneering work of Jeffery (1922), who obtained an equation for the motion of an ellipsoidal particle immersed in a Newtonian fluid, despite the fact that this model relies on strong assumptions: negligible inertia, unconfined flow, dilute regime, flow unperturbed by the presence of the suspended particle, etc. In this work, we propose a dumbbellbased model aimed to describe the motion of an inertial fibre or ellipsoid suspended in a Newtonian fluid. We then use this model to study the orientation kinematics of such particle in a linear shear flow and compare it to the inertialess case. In the case of fibres, we observe the appearance of periodic orbits (whereas inertialess fibres just align in the flow field). For spheroids, our model predicts an orbit drift towards the flowgradient plane, either gradually (slight inertia) or by first rotating around a moving oblique axis (heavy particles). MultiParticle Collision Dynamics (MPCD) simulations were carried out to assess the model predictions in the case of inertial fibres and revealed similar behaviours.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): A. Scheuer, G. Grégoire, E. AbissetChavanne, F. Chinesta, R. KeuningsAbstractSimulations of flows containing nonspherical particles (fibres or ellipsoids) rely on the knowledge of the equation governing the particle motion in the flow. Most models used nowadays are based on the pioneering work of Jeffery (1922), who obtained an equation for the motion of an ellipsoidal particle immersed in a Newtonian fluid, despite the fact that this model relies on strong assumptions: negligible inertia, unconfined flow, dilute regime, flow unperturbed by the presence of the suspended particle, etc. In this work, we propose a dumbbellbased model aimed to describe the motion of an inertial fibre or ellipsoid suspended in a Newtonian fluid. We then use this model to study the orientation kinematics of such particle in a linear shear flow and compare it to the inertialess case. In the case of fibres, we observe the appearance of periodic orbits (whereas inertialess fibres just align in the flow field). For spheroids, our model predicts an orbit drift towards the flowgradient plane, either gradually (slight inertia) or by first rotating around a moving oblique axis (heavy particles). MultiParticle Collision Dynamics (MPCD) simulations were carried out to assess the model predictions in the case of inertial fibres and revealed similar behaviours.

H 3
solutions in H 3 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): J.A. Ferreira, P. de Oliveira, E. SilveiraAbstractIn this paper we consider the coupling between two quasilinear diffusion equations: the diffusion coefficient of the first equation depends on its solution and the diffusion and convective coefficients of the second equation depend on the solution of the first one. This system can be used to describe the drug evolution in a target tissue when the drug transport is enhanced by heat. We study, from an analytical and a numerical viewpoints, the coupling of the heat equation with the drug diffusion equation. A fully discrete piecewise linear finite method is proposed to solve this system and its stability is studied. Assuming that the heat and the concentration are in H3 we prove that the method is second order convergent. Numerical experiments illustrating the theoretical results and the global qualitative behaviour of the coupling are also included.
 Abstract: Publication date: 1 February 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 3Author(s): J.A. Ferreira, P. de Oliveira, E. SilveiraAbstractIn this paper we consider the coupling between two quasilinear diffusion equations: the diffusion coefficient of the first equation depends on its solution and the diffusion and convective coefficients of the second equation depend on the solution of the first one. This system can be used to describe the drug evolution in a target tissue when the drug transport is enhanced by heat. We study, from an analytical and a numerical viewpoints, the coupling of the heat equation with the drug diffusion equation. A fully discrete piecewise linear finite method is proposed to solve this system and its stability is studied. Assuming that the heat and the concentration are in H3 we prove that the method is second order convergent. Numerical experiments illustrating the theoretical results and the global qualitative behaviour of the coupling are also included.
 A simple, highorder and compact WENO limiter for RKDG method
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Hongqiang Zhu, Jianxian Qiu, Jun ZhuAbstractIn this paper, a new limiter using weighted essentially nonoscillatory (WENO) methodology is investigated for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving hyperbolic conservation laws. The idea is to use the highorder DG solution polynomial itself in the target cell and the linear polynomials which are reconstructed by the cell averages of solution in the target cell and its neighboring cells to reconstruct a new highorder polynomial in a manner of WENO methodology. Since only the linear polynomials need to be prepared for reconstruction, this limiter is very simple and compact with a stencil including only the target cell and its immediate neighboring cells. Numerical examples of various problems show that the new limiting procedure can simultaneously achieve uniform highorder accuracy and sharp, nonoscillatory shock transitions.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Hongqiang Zhu, Jianxian Qiu, Jun ZhuAbstractIn this paper, a new limiter using weighted essentially nonoscillatory (WENO) methodology is investigated for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving hyperbolic conservation laws. The idea is to use the highorder DG solution polynomial itself in the target cell and the linear polynomials which are reconstructed by the cell averages of solution in the target cell and its neighboring cells to reconstruct a new highorder polynomial in a manner of WENO methodology. Since only the linear polynomials need to be prepared for reconstruction, this limiter is very simple and compact with a stencil including only the target cell and its immediate neighboring cells. Numerical examples of various problems show that the new limiting procedure can simultaneously achieve uniform highorder accuracy and sharp, nonoscillatory shock transitions.
 Multidimensional spectral tau methods for distributedorder fractional
diffusion equations Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Mahmoud A. Zaky, J. Tenreiro MachadoAbstractThe distributedorder fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking powerlaw scaling over the whole timedomain. An important application of distributedorder diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate. To broaden the range of applicability of distributedorder fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we develop spectral tau schemes to discretize the fractional diffusion equation with distributedorder fractional derivative in time and Dirichlet boundary conditions. The model solution is expanded in multidimensions in terms of Legendre polynomials and the discrete equations are obtained with the tau method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth. Our results confirm that nonlocal numerical methods are best suited to discretize distributedorder fractional differential equations as they naturally take the global behavior of the solution into account.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Mahmoud A. Zaky, J. Tenreiro MachadoAbstractThe distributedorder fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking powerlaw scaling over the whole timedomain. An important application of distributedorder diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate. To broaden the range of applicability of distributedorder fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we develop spectral tau schemes to discretize the fractional diffusion equation with distributedorder fractional derivative in time and Dirichlet boundary conditions. The model solution is expanded in multidimensions in terms of Legendre polynomials and the discrete equations are obtained with the tau method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth. Our results confirm that nonlocal numerical methods are best suited to discretize distributedorder fractional differential equations as they naturally take the global behavior of the solution into account.
 Limiting dynamics for stochastic reaction–diffusion equations on the
Sobolev space with thin domains Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Fuzhi Li, Yangrong Li, Renhai WangAbstractThis paper is devoted to bispatial random attractors of the stochastic reaction–diffusion equation when the terminate space is the Sobolev space on a thin domain, where the nonlinearity can be decomposed into two functions with (p,q)growth exponents. By means of a computation method of induction, it is shown that the difference of solutions near the initial time is integrable of kp−2k+2 order. This higherorder integrability shows continuity of the solution operator from the square Lebesgue space to (kp−2k+2)order Lebesgue spaces. In particular, the (2p−2)order integrability shows continuity of the solution operator from the square Lebesgue space to the Sobolev space, which further shows existence of a random attractor in the Sobolev space when the initial space is the square Lebesgue space. Moreover, the higherorder integrability can be uniform with respect to all thin domains, which provides uniformly asymptotic compactness of the random dynamical systems. As a conclusion, the upper semicontinuity of attractors under the Sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Fuzhi Li, Yangrong Li, Renhai WangAbstractThis paper is devoted to bispatial random attractors of the stochastic reaction–diffusion equation when the terminate space is the Sobolev space on a thin domain, where the nonlinearity can be decomposed into two functions with (p,q)growth exponents. By means of a computation method of induction, it is shown that the difference of solutions near the initial time is integrable of kp−2k+2 order. This higherorder integrability shows continuity of the solution operator from the square Lebesgue space to (kp−2k+2)order Lebesgue spaces. In particular, the (2p−2)order integrability shows continuity of the solution operator from the square Lebesgue space to the Sobolev space, which further shows existence of a random attractor in the Sobolev space when the initial space is the square Lebesgue space. Moreover, the higherorder integrability can be uniform with respect to all thin domains, which provides uniformly asymptotic compactness of the random dynamical systems. As a conclusion, the upper semicontinuity of attractors under the Sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain.
 A fast preconditioned iterative method for twodimensional options pricing
under fractional differential models Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Xu Chen, Deng Ding, SiuLong Lei, Wenfei WangAbstractIn recent years, fractional partial differential equation (FPDE) has been widely applied in options pricing problems, which better explains many important empirical facts of financial markets. However, the vast majority of the literatures focus on pricing the single asset option under the FPDE framework. In this paper, a twodimensional FPDE governing the valuation of rainbow options is established, where two underlying assets are assumed to follow independent exponential Lévy processes, and its boundary conditions are determined by solving onedimensional FPDEs. A secondorder accurate finite difference scheme is proposed to discretize the twodimensional FPDE. Given the block Toeplitz with Toeplitz block structure of the coefficient matrix, a fast preconditioned Krylov subspace method is developed for solving the resulting linear system with O(NlogN) computational complexity per iteration, where N is the matrix size. The proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Xu Chen, Deng Ding, SiuLong Lei, Wenfei WangAbstractIn recent years, fractional partial differential equation (FPDE) has been widely applied in options pricing problems, which better explains many important empirical facts of financial markets. However, the vast majority of the literatures focus on pricing the single asset option under the FPDE framework. In this paper, a twodimensional FPDE governing the valuation of rainbow options is established, where two underlying assets are assumed to follow independent exponential Lévy processes, and its boundary conditions are determined by solving onedimensional FPDEs. A secondorder accurate finite difference scheme is proposed to discretize the twodimensional FPDE. Given the block Toeplitz with Toeplitz block structure of the coefficient matrix, a fast preconditioned Krylov subspace method is developed for solving the resulting linear system with O(NlogN) computational complexity per iteration, where N is the matrix size. The proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods.
 An RBFFD sparse scheme to simulate highdimensional Black–Scholes
partial differential equations Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Malik Zaka UllahAbstractThe objective of this research is to investigate the numerical solution of highdimensional Black–Scholes partial differential equations (PDEs). To do this, the weights of the radial basis functionfinite difference (RBFFD) scheme using inverse multiquadric function are used with an emphasis on the hotzone. The proposed RBFFD technique is capable to price multiasset options efficiently.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Malik Zaka UllahAbstractThe objective of this research is to investigate the numerical solution of highdimensional Black–Scholes partial differential equations (PDEs). To do this, the weights of the radial basis functionfinite difference (RBFFD) scheme using inverse multiquadric function are used with an emphasis on the hotzone. The proposed RBFFD technique is capable to price multiasset options efficiently.
 Lowrank and sparse matrices fitting algorithm for lowrank representation
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Jianxi Zhao, Lina ZhaoAbstractIn real world, especially in the field of pattern recognition, a matrix formed from images, visions, speech sounds or so forth under certain conditions usually subjects to a lowrank subspace structure. Sparse noise, small noise and so on can be eliminated by the lowrank property of this matrix, leading to the wellknown lowrank representation problem. At present, existing algorithms for this problem still need to be improved in the aspects of the recovery accuracy of lowrank component and sparse component, the clustering accuracy of subspaces’ data and their convergence rate. This paper proposes a lowrank matrix decomposition nonconvex optimization extended model without nuclear norm. Motivated by human walking, we combine the direction and step size iterative formula with the alternating direction minimization idea for the sake of decomposing the original optimization model that is difficult to be solved into three comparatively easily solved suboptimization models. On the basis of these, LowRank and Sparse Matrices Fitting Algorithm (LSMF) is presented for the submodels in this paper, which quickly alternates the search direction matrices and the corresponding step sizes. Theoretically, it is proved that LSMF converges to a stable point of the extended model. In simulation experiments, better results are achieved in the three aspects under appropriate conditions. The face denoising and background/foreground separation further demonstrate the capability of LSMF on handling largescale and contaminated dataset.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Jianxi Zhao, Lina ZhaoAbstractIn real world, especially in the field of pattern recognition, a matrix formed from images, visions, speech sounds or so forth under certain conditions usually subjects to a lowrank subspace structure. Sparse noise, small noise and so on can be eliminated by the lowrank property of this matrix, leading to the wellknown lowrank representation problem. At present, existing algorithms for this problem still need to be improved in the aspects of the recovery accuracy of lowrank component and sparse component, the clustering accuracy of subspaces’ data and their convergence rate. This paper proposes a lowrank matrix decomposition nonconvex optimization extended model without nuclear norm. Motivated by human walking, we combine the direction and step size iterative formula with the alternating direction minimization idea for the sake of decomposing the original optimization model that is difficult to be solved into three comparatively easily solved suboptimization models. On the basis of these, LowRank and Sparse Matrices Fitting Algorithm (LSMF) is presented for the submodels in this paper, which quickly alternates the search direction matrices and the corresponding step sizes. Theoretically, it is proved that LSMF converges to a stable point of the extended model. In simulation experiments, better results are achieved in the three aspects under appropriate conditions. The face denoising and background/foreground separation further demonstrate the capability of LSMF on handling largescale and contaminated dataset.
 A class of secondorder McKean–Vlasov stochastic evolution equations
driven by fractional Brownian motion and Poisson jumps Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Mark A. McKibben, Micah WebsterAbstractThis paper focuses on a class of secondorder McKean–Vlasov stochastic evolution equations driven by a fractional Brownian motion and Poisson jumps. Specifically, we allow nonlinearities and the jump term to depend not only of the state of the solution, but also on the corresponding probability law of the state. The global existence and uniqueness of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Mark A. McKibben, Micah WebsterAbstractThis paper focuses on a class of secondorder McKean–Vlasov stochastic evolution equations driven by a fractional Brownian motion and Poisson jumps. Specifically, we allow nonlinearities and the jump term to depend not only of the state of the solution, but also on the corresponding probability law of the state. The global existence and uniqueness of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory.
 A nonlocal reaction–diffusion prey–predator model with free
boundary Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Chenglin LiAbstractThis paper is concerned with a nonlocal reaction–diffusion prey–predator model with free boundary. The spatial–temporal risk index R0p(t) which is threshold value, is introduced for this model. The global existence of positive solutions is proved and the longtime behavior of positive solutions is investigated. The sufficient conditions are found for vanishing of predators and the establishment of themselves in new environment. Furthermore, the spreading–vanishing criteria are established. The results show that the predators will vanish finally and not establish themselves if the sizes of habitant and spreading coefficient are relatively small or the initial value of predators is also relatively small with R0p(0)1 for t0>0.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Chenglin LiAbstractThis paper is concerned with a nonlocal reaction–diffusion prey–predator model with free boundary. The spatial–temporal risk index R0p(t) which is threshold value, is introduced for this model. The global existence of positive solutions is proved and the longtime behavior of positive solutions is investigated. The sufficient conditions are found for vanishing of predators and the establishment of themselves in new environment. Furthermore, the spreading–vanishing criteria are established. The results show that the predators will vanish finally and not establish themselves if the sizes of habitant and spreading coefficient are relatively small or the initial value of predators is also relatively small with R0p(0)1 for t0>0.
 Variational multiscale interpolating elementfree Galerkin method for the
nonlinear Darcy–Forchheimer model Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Tao Zhang, Xiaolin LiAbstractIn this paper, the variational multiscale interpolating elementfree Galerkin (VMIEFG) method is developed to obtain the numerical solution of the nonlinear Darcy–Forchheimer model. We use the interpolating moving least squares method instead of the moving least squares approximation to construct meshless shape functions with delta function properties. Then the flux boundary condition of the Darcy–Forchheimer model can be handled easily. Hughes’ variational multiscale (HVM) method is applied to overcome the numerical oscillation caused by equalorder basis for the velocity and pressure. Moreover, the HVM ensures that the resultant formulation in the VMIEFG method is consistent and the stabilization parameter (or tensor) appears naturally. Consequently, the stabilization parameter is free of userdefined. The fixed point iteration method is used to deal with the nonlinear term. Some numerical examples are provided to illustrate the stability and performance of the proposed method for solving the nonlinear Darcy–Forchheimer model.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Tao Zhang, Xiaolin LiAbstractIn this paper, the variational multiscale interpolating elementfree Galerkin (VMIEFG) method is developed to obtain the numerical solution of the nonlinear Darcy–Forchheimer model. We use the interpolating moving least squares method instead of the moving least squares approximation to construct meshless shape functions with delta function properties. Then the flux boundary condition of the Darcy–Forchheimer model can be handled easily. Hughes’ variational multiscale (HVM) method is applied to overcome the numerical oscillation caused by equalorder basis for the velocity and pressure. Moreover, the HVM ensures that the resultant formulation in the VMIEFG method is consistent and the stabilization parameter (or tensor) appears naturally. Consequently, the stabilization parameter is free of userdefined. The fixed point iteration method is used to deal with the nonlinear term. Some numerical examples are provided to illustrate the stability and performance of the proposed method for solving the nonlinear Darcy–Forchheimer model.
 The efficient rotational pressurecorrection schemes for the coupling
Stokes/Darcy problem Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Jian Li, Min Yao, Md. Abdullah Al Mahbub, Haibiao ZhengAbstractIn this paper, the rotational pressurecorrection methods for the Stokes/Darcy system are developed and analyzed. The central advantage of these methods is a timedependent version of domain decomposition. These methods have firstorder/secondorder accuracy without the incompressibility constraint of the Stokes/Darcy system. Their main feature is the implementation efficiency in that we only solve one vectorvalued elliptic equation and one scalarvalued Poisson equation for the Stokes equations per time step. The unconditional stability and long time stability are established and numerical experiments are also presented to show their performance.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Jian Li, Min Yao, Md. Abdullah Al Mahbub, Haibiao ZhengAbstractIn this paper, the rotational pressurecorrection methods for the Stokes/Darcy system are developed and analyzed. The central advantage of these methods is a timedependent version of domain decomposition. These methods have firstorder/secondorder accuracy without the incompressibility constraint of the Stokes/Darcy system. Their main feature is the implementation efficiency in that we only solve one vectorvalued elliptic equation and one scalarvalued Poisson equation for the Stokes equations per time step. The unconditional stability and long time stability are established and numerical experiments are also presented to show their performance.

R N
concave–convex nonlinearities in R N Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): DongLun Wu, Fengying LiAbstractIn this paper, we show the existence and multiplicity of solutions for the following fourthorder Kirchhoff type elliptic equations Δ2u−M(‖∇u‖22)Δu+V(x)u=f(x,u),x∈RN,where M(t):R→R is the Kirchhoff function, f(x,u)=λk(x,u)+h(x,u), λ≥0, k(x,u) is of sublinear growth and h(x,u) satisfies some general 3superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for λ=0. For λ>0 small enough, we obtain at least two nontrivial solutions. Furthermore, if f(x,u) is odd in u, we show that above equations possess infinitely many solutions for all λ≥0. Our theorems generalize some known results in the literatures even for λ=0 and our proof is based on the variational methods.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): DongLun Wu, Fengying LiAbstractIn this paper, we show the existence and multiplicity of solutions for the following fourthorder Kirchhoff type elliptic equations Δ2u−M(‖∇u‖22)Δu+V(x)u=f(x,u),x∈RN,where M(t):R→R is the Kirchhoff function, f(x,u)=λk(x,u)+h(x,u), λ≥0, k(x,u) is of sublinear growth and h(x,u) satisfies some general 3superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for λ=0. For λ>0 small enough, we obtain at least two nontrivial solutions. Furthermore, if f(x,u) is odd in u, we show that above equations possess infinitely many solutions for all λ≥0. Our theorems generalize some known results in the literatures even for λ=0 and our proof is based on the variational methods.
 Orbital stability of standing waves for Schrödinger type equations with
slowly decaying linear potential Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Xinfu Li, Junying ZhaoAbstractIn this paper, kinds of Schrödinger type equations with slowly decaying linear potential and power type or convolution type nonlinearities are considered. By using the concentration compactness principle, the sharp Gagliardo–Nirenberg inequality and a refined estimate of the linear operator, the existence and orbital stability of standing waves in L2subcritical and L2critical cases are established in a systematic way.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Xinfu Li, Junying ZhaoAbstractIn this paper, kinds of Schrödinger type equations with slowly decaying linear potential and power type or convolution type nonlinearities are considered. By using the concentration compactness principle, the sharp Gagliardo–Nirenberg inequality and a refined estimate of the linear operator, the existence and orbital stability of standing waves in L2subcritical and L2critical cases are established in a systematic way.
 A modified quasiboundary value method for a twodimensional inverse heat
conduction problem Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Wei Cheng, Qi ZhaoAbstractIn this study, we consider a twodimensional inverse heat conduction problem, which determines the surface temperature and heat flux distribution from measured data at the fixed location. The problem is seriously ill posed in the Hadamard sense and a conditional stability is given for it. We propose a modified quasiboundary value regularization method to deal with the illposed problem. By choosing suitable regularization parameters and introducing some technical inequalities, we obtain quite sharp error estimates between the approximate solution and its exact solution.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Wei Cheng, Qi ZhaoAbstractIn this study, we consider a twodimensional inverse heat conduction problem, which determines the surface temperature and heat flux distribution from measured data at the fixed location. The problem is seriously ill posed in the Hadamard sense and a conditional stability is given for it. We propose a modified quasiboundary value regularization method to deal with the illposed problem. By choosing suitable regularization parameters and introducing some technical inequalities, we obtain quite sharp error estimates between the approximate solution and its exact solution.
 A novel numerical method for accelerating the computation of the
steadystate in induction machines Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): A. Bermúdez, D. Gómez, M. Piñeiro, P. SalgadoAbstractThis paper presents a novel and efficient methodology to reduce the time needed to reach the steadystate in the finite element simulation of induction machines. More precisely, the work focuses on induction motors with squirrel cage rotor, where sources in the stator coil sides are given in terms of periodic currents. Essentially, the procedure consists in computing suitable initial conditions for the currents in the rotor bars, thus allowing to obtain the steadystate fields of the machine by solving a transient magnetic model in just a few revolutions. Firstly, the mathematical model that simulates the behavior of the machine is introduced. Then, an approximation of this model is developed, from which suitable initial currents are derived by computing the solution in the leastsquare sense to an overdetermined problem with only two unknowns. Finally, the method is applied to a particular induction machine working under different operating conditions. The results show important computational savings to reach the motor steadystate in comparison with assuming zero initial conditions, which validate the efficiency of the procedure.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): A. Bermúdez, D. Gómez, M. Piñeiro, P. SalgadoAbstractThis paper presents a novel and efficient methodology to reduce the time needed to reach the steadystate in the finite element simulation of induction machines. More precisely, the work focuses on induction motors with squirrel cage rotor, where sources in the stator coil sides are given in terms of periodic currents. Essentially, the procedure consists in computing suitable initial conditions for the currents in the rotor bars, thus allowing to obtain the steadystate fields of the machine by solving a transient magnetic model in just a few revolutions. Firstly, the mathematical model that simulates the behavior of the machine is introduced. Then, an approximation of this model is developed, from which suitable initial currents are derived by computing the solution in the leastsquare sense to an overdetermined problem with only two unknowns. Finally, the method is applied to a particular induction machine working under different operating conditions. The results show important computational savings to reach the motor steadystate in comparison with assuming zero initial conditions, which validate the efficiency of the procedure.
 Alternating direction implicit difference scheme for the multiterm
timefractional integrodifferential equation with a weakly singular
kernel Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Jun Zhou, Da XuAbstractWe present an alternating direction implicit (ADI) difference scheme for the multiterm timefractional integrodifferential equation with a weakly singular kernel. The Caputo fractional derivative is discretized by L1 method and the integrodifferential term is discretized by a convolution quadrature suggested by Lubich. The stability and optimal convergence order estimate of the difference scheme are derived. Numerical experiments verify our theory analysis.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Jun Zhou, Da XuAbstractWe present an alternating direction implicit (ADI) difference scheme for the multiterm timefractional integrodifferential equation with a weakly singular kernel. The Caputo fractional derivative is discretized by L1 method and the integrodifferential term is discretized by a convolution quadrature suggested by Lubich. The stability and optimal convergence order estimate of the difference scheme are derived. Numerical experiments verify our theory analysis.
 Superconvergent analysis of a nonconforming mixed finite element method
for nonstationary conduction–convection problem Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Dongyang Shi, Qian LiuAbstractIn this paper, a backwardEuler mixed finite element (MFE) approximation scheme is presented for the nonstationary conduction–convection problem, in which the constrained nonconforming rotated (CNR) Q1 element is used to approximate the velocity u, the temperature T and the Q0 element to the pressure p. Based on the characters of these elements and some special skills, i.e., meanvalue skill and a new transforming skill with respect to τ, the superclose results of order O(h2+τ) for u,T in the broken H1norm and p in L2norm are deduced, respectively. Here, h is the subdivision parameter and τ, the time step. Furthermore, the global superconvergent results are obtained through the interpolation postprocessing technique. Finally, numerical results are provided to confirm the theoretical analysis.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): Dongyang Shi, Qian LiuAbstractIn this paper, a backwardEuler mixed finite element (MFE) approximation scheme is presented for the nonstationary conduction–convection problem, in which the constrained nonconforming rotated (CNR) Q1 element is used to approximate the velocity u, the temperature T and the Q0 element to the pressure p. Based on the characters of these elements and some special skills, i.e., meanvalue skill and a new transforming skill with respect to τ, the superclose results of order O(h2+τ) for u,T in the broken H1norm and p in L2norm are deduced, respectively. Here, h is the subdivision parameter and τ, the time step. Furthermore, the global superconvergent results are obtained through the interpolation postprocessing technique. Finally, numerical results are provided to confirm the theoretical analysis.
 Isogeometric Residual Minimization Method (iGRM) with direction splitting
for nonstationary advection–diffusion problems Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): M. Łoś, J. MuñozMatute, I. Muga, M. PaszyńskiAbstractIn this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product Bspline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integration scheme and apply, for each spacedirection, a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method on three advection–diffusion computational examples, including model “membrane” problem, the circular wind problem, and the simulations modeling pollution propagating from a chimney.
 Abstract: Publication date: 15 January 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 2Author(s): M. Łoś, J. MuñozMatute, I. Muga, M. PaszyńskiAbstractIn this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product Bspline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integration scheme and apply, for each spacedirection, a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method on three advection–diffusion computational examples, including model “membrane” problem, the circular wind problem, and the simulations modeling pollution propagating from a chimney.
 Method of analytical regularization in the analysis of axially symmetric
excitation of imperfect circular disk antennas Abstract: Publication date: Available online 2 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Nataliya Y. Saidoglu, Alexander I. NosichAbstractThis work presents accurate numerical method and computational results in the modeling of the axisymmetric radiation of resistive and thindielectric circular diskonsubstrate antennas. To simulate resistive and thindielectric disk features, we use twoside Generalized Boundary Conditions (GBC). The analysis is based on the singular integral equations (IEs) and the Method of Analytical Regularization (MAR) exploiting a Galerkin Method (GM) with judicious basis functions, which convert IEs into the Fredholm secondkind infinite matrix equations. Unlike previously known numerical techniques, this guarantees convergence and enables us to compute the solution with controlled accuracy even in sharp resonances. Numerical results demonstrate the effect of disk losses and transparency on the fundamental radiation phenomena and help improve antenna characteristics such as radiation efficiency.
 Abstract: Publication date: Available online 2 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Nataliya Y. Saidoglu, Alexander I. NosichAbstractThis work presents accurate numerical method and computational results in the modeling of the axisymmetric radiation of resistive and thindielectric circular diskonsubstrate antennas. To simulate resistive and thindielectric disk features, we use twoside Generalized Boundary Conditions (GBC). The analysis is based on the singular integral equations (IEs) and the Method of Analytical Regularization (MAR) exploiting a Galerkin Method (GM) with judicious basis functions, which convert IEs into the Fredholm secondkind infinite matrix equations. Unlike previously known numerical techniques, this guarantees convergence and enables us to compute the solution with controlled accuracy even in sharp resonances. Numerical results demonstrate the effect of disk losses and transparency on the fundamental radiation phenomena and help improve antenna characteristics such as radiation efficiency.
 Solving quantum stochastic LQR optimal control problem in Fock space and
its application in finance Abstract: Publication date: Available online 31 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): S. Yaghobipour, M. YarahmadiAbstractThis paper is an attempt for solving operatorvalued quantum stochastic optimal control problems, in Fock space. For this purpose, the dynamics of the classical system state is described by Hudson–Parthasarathy type Quantum Stochastic Differential Equation (QSDE) in Fock space and then by associating a quadratic performance criterion with the QSDE, a Quantum Stochastic Linear Quadratic Regulator (QSLQR) optimal control problem is formulated. Also, an algorithm for solving the QSLQR optimal control problem is designed. For solving the resulting optimal control problem, a new HJB equation is obtained. Thereby, the operator valued control process is obtained. Two theorems are proved to facilitate the algorithm. In this paper, for the first time, the optimal strategy for trading stock is designed via the presented method. For this purpose, Merton portfolio allocation problem is solved. The simulation results show that portfolio optimal performances, minimum risk and maximum return are achieved via presented method.
 Abstract: Publication date: Available online 31 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): S. Yaghobipour, M. YarahmadiAbstractThis paper is an attempt for solving operatorvalued quantum stochastic optimal control problems, in Fock space. For this purpose, the dynamics of the classical system state is described by Hudson–Parthasarathy type Quantum Stochastic Differential Equation (QSDE) in Fock space and then by associating a quadratic performance criterion with the QSDE, a Quantum Stochastic Linear Quadratic Regulator (QSLQR) optimal control problem is formulated. Also, an algorithm for solving the QSLQR optimal control problem is designed. For solving the resulting optimal control problem, a new HJB equation is obtained. Thereby, the operator valued control process is obtained. Two theorems are proved to facilitate the algorithm. In this paper, for the first time, the optimal strategy for trading stock is designed via the presented method. For this purpose, Merton portfolio allocation problem is solved. The simulation results show that portfolio optimal performances, minimum risk and maximum return are achieved via presented method.
 Boundary integral approach for the mixed Dirichlet–Robin boundary value
problem for the nonlinear Darcy–Forchheimer–Brinkman system Abstract: Publication date: Available online 30 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Robert GuttAbstractThe purpose of this paper is to provide the solvability and uniqueness result to the mixed Dirichlet–Robin boundary value problem for the nonlinear Darcy–Forchheimer–Brinkman system in a bounded, twodimensional Lipschitz domain. First we obtain a wellposedness result for the linear Brinkman system with Dirichlet–Neumann boundary conditions, by reducing the problem to the system of boundary integral equations based on the fundamental solution of the Brinkman system and by analyzing this system employing a variational approach. The result is extended afterwards to the Poisson problem for the Brinkman system and to Dirichlet–Robin boundary conditions, using the Newtonian potential and the linearity of the solution operator. Further, we study the nonlinear Darcy–Forchheimer–Brinkman boundary value problem of Dirichlet and Robin type.
 Abstract: Publication date: Available online 30 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Robert GuttAbstractThe purpose of this paper is to provide the solvability and uniqueness result to the mixed Dirichlet–Robin boundary value problem for the nonlinear Darcy–Forchheimer–Brinkman system in a bounded, twodimensional Lipschitz domain. First we obtain a wellposedness result for the linear Brinkman system with Dirichlet–Neumann boundary conditions, by reducing the problem to the system of boundary integral equations based on the fundamental solution of the Brinkman system and by analyzing this system employing a variational approach. The result is extended afterwards to the Poisson problem for the Brinkman system and to Dirichlet–Robin boundary conditions, using the Newtonian potential and the linearity of the solution operator. Further, we study the nonlinear Darcy–Forchheimer–Brinkman boundary value problem of Dirichlet and Robin type.
 Mean escape time for randomly switching narrow gates in a steady flow
 Abstract: Publication date: Available online 28 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hui Wang, Jinqiao Duan, Xianguo Geng, Ying ChaoAbstractThe escape of particles through a narrow absorbing gate in confined domains is a rich phenomenon in various systems in physics, chemistry and molecular biophysics. We consider the narrow escape problem in a bounded annular when the two gates randomly switch between different states with a switching rate k between the two gates. After briefly deriving the coupled partial differential equations for the escape time through two gates, we compute the mean escape time for particles escaping from the gates with different initial states. By numerical simulation under nonuniform boundary conditions, we quantify how narrow escape time is affected by the switching rate k between the two gates, arc length s between two gates, angular velocity w of the steady flow and diffusion coefficient D. We reveal that the mean escape time decreases with the switching rate k between the two gates, angular velocity w and diffusion coefficient D for fixed arc length, but takes the minimum when the two gates are evenly separated on the boundary for any given switching rate k between the two gates. In particular, we find that when the arc length size ε for the gates is sufficiently small, the average narrow escape time is approximately independent of the gate arc length size. We further indicate combinations of system parameters (regions located in the parameter space) such that the mean escape time is the longest or shortest. Our findings provide mathematical understanding for phenomena such as how ions select ion channels and how chemicals leak in annulus ring containers, when drift vector fields are present.
 Abstract: Publication date: Available online 28 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hui Wang, Jinqiao Duan, Xianguo Geng, Ying ChaoAbstractThe escape of particles through a narrow absorbing gate in confined domains is a rich phenomenon in various systems in physics, chemistry and molecular biophysics. We consider the narrow escape problem in a bounded annular when the two gates randomly switch between different states with a switching rate k between the two gates. After briefly deriving the coupled partial differential equations for the escape time through two gates, we compute the mean escape time for particles escaping from the gates with different initial states. By numerical simulation under nonuniform boundary conditions, we quantify how narrow escape time is affected by the switching rate k between the two gates, arc length s between two gates, angular velocity w of the steady flow and diffusion coefficient D. We reveal that the mean escape time decreases with the switching rate k between the two gates, angular velocity w and diffusion coefficient D for fixed arc length, but takes the minimum when the two gates are evenly separated on the boundary for any given switching rate k between the two gates. In particular, we find that when the arc length size ε for the gates is sufficiently small, the average narrow escape time is approximately independent of the gate arc length size. We further indicate combinations of system parameters (regions located in the parameter space) such that the mean escape time is the longest or shortest. Our findings provide mathematical understanding for phenomena such as how ions select ion channels and how chemicals leak in annulus ring containers, when drift vector fields are present.
 A robust sharp interface based immersed boundary framework for moving body
problems with applications to laminar incompressible flows Abstract: Publication date: Available online 27 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Pradeep Kumar Seshadri, Ashoke DeAbstractThis study presents a robust, sharp interface immersed boundary (IBM) framework for moving body problems. The inhouse solver makes use of a densitybased finite volume framework for solving unsteady, 3D Favre averaged Navier Stokes equation in a generalized curvilinear coordinate system. The immersed boundary formulation is capable of handling arbitrarily complex threedimensional bodies. The sharp interface approach allows for the exact imposition of boundary conditions at the immersed surface by reconstructing the flow field along its local normal. The implemented reconstruction schemes maintain secondorder accuracy. The study focuses on issues of mass conservation and spurious temporal oscillations (pressure as well as force) that the sharp interface IBM approach typically faces when encountering moving body problems. A ghost nodebased field extension technique provides an efficient way to improve mass conservation and as a result, reduces not only spurious oscillations but also increases temporal accuracy. Flow past both bluff body (triangular, circular and spherical), as well as streamlined body (airfoil), is presented here as validation studies. The ability of the present formulation to deal with moving body problems in the laminar incompressible flow regime is demonstrated by presenting cases that involve motions such as pitching and inline oscillation. The predictions are found to be in good agreement with the published results and measurements as well.
 Abstract: Publication date: Available online 27 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Pradeep Kumar Seshadri, Ashoke DeAbstractThis study presents a robust, sharp interface immersed boundary (IBM) framework for moving body problems. The inhouse solver makes use of a densitybased finite volume framework for solving unsteady, 3D Favre averaged Navier Stokes equation in a generalized curvilinear coordinate system. The immersed boundary formulation is capable of handling arbitrarily complex threedimensional bodies. The sharp interface approach allows for the exact imposition of boundary conditions at the immersed surface by reconstructing the flow field along its local normal. The implemented reconstruction schemes maintain secondorder accuracy. The study focuses on issues of mass conservation and spurious temporal oscillations (pressure as well as force) that the sharp interface IBM approach typically faces when encountering moving body problems. A ghost nodebased field extension technique provides an efficient way to improve mass conservation and as a result, reduces not only spurious oscillations but also increases temporal accuracy. Flow past both bluff body (triangular, circular and spherical), as well as streamlined body (airfoil), is presented here as validation studies. The ability of the present formulation to deal with moving body problems in the laminar incompressible flow regime is demonstrated by presenting cases that involve motions such as pitching and inline oscillation. The predictions are found to be in good agreement with the published results and measurements as well.
 An optimal error estimate for the twodimensional nonlinear time
fractional advection–diffusion equation with smooth and nonsmooth
solutions Abstract: Publication date: Available online 26 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hui Zhang, Junqing Jia, Xiaoyun JiangAbstractIn this paper, a linearized L1 Legendre–Galerkin spectral method for the twodimensional nonlinear time fractional advection–diffusion equation is derived. The numerical method is stable without the CFL conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. We also consider the case of nonsmooth solutions by adding some correction terms. The numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method.
 Abstract: Publication date: Available online 26 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hui Zhang, Junqing Jia, Xiaoyun JiangAbstractIn this paper, a linearized L1 Legendre–Galerkin spectral method for the twodimensional nonlinear time fractional advection–diffusion equation is derived. The numerical method is stable without the CFL conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. We also consider the case of nonsmooth solutions by adding some correction terms. The numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method.
 Two improved electronegativity equalization methods for charge
distribution in large scale nonuniform system Abstract: Publication date: Available online 26 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hanqing Li, Bonan Xu, Hanhui Jin, Haiou Wang, Jianren FanAbstractNowadays, molecular dynamics (MD) using force fields has become an effective tool for physical and chemical problems. In the MD method, the classical Electronegativity Equalization Method (EEM) is widely accepted for atom charge computing, which may result in unreasonable predictions for large scale nonuniform systems. In order to improve the accuracy of charge computing in MD simulations, two versions of improved EEM for charge distribution calculation are proposed in the present work, without introducing new empirical parameters, namely the locally equilibrated EEM and the subzone EEM. The aim of the proposed methods is to eliminate the unreasonable interaction of the longrange charge in large scale nonuniform systems. Comparison between the two improved EEMs and the original EEM in both uniform and nonuniform systems shows that both the improved EEMs successfully eliminate the effect of the unreasonable longrange charge interaction which is inevitably encountered in the original EEM.
 Abstract: Publication date: Available online 26 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Hanqing Li, Bonan Xu, Hanhui Jin, Haiou Wang, Jianren FanAbstractNowadays, molecular dynamics (MD) using force fields has become an effective tool for physical and chemical problems. In the MD method, the classical Electronegativity Equalization Method (EEM) is widely accepted for atom charge computing, which may result in unreasonable predictions for large scale nonuniform systems. In order to improve the accuracy of charge computing in MD simulations, two versions of improved EEM for charge distribution calculation are proposed in the present work, without introducing new empirical parameters, namely the locally equilibrated EEM and the subzone EEM. The aim of the proposed methods is to eliminate the unreasonable interaction of the longrange charge in large scale nonuniform systems. Comparison between the two improved EEMs and the original EEM in both uniform and nonuniform systems shows that both the improved EEMs successfully eliminate the effect of the unreasonable longrange charge interaction which is inevitably encountered in the original EEM.
 Flows of real gas in nozzles with unsteady local energy supply
 Abstract: Publication date: Available online 24 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): N.A. Brykov, V.N. Emelyanov, A.G. Karpenko, K.N. VolkovAbstractWhen gas flows at a high speed in a channel with a variable cross sectional area and highintensity energy supply, it experiences complicated physical and chemical processes producing hightemperature gas effects. Hightemperature gas effects are a key issue related to design and optimization of nozzles of plasmatron of alternating current. The finite volume method is applied to solve unsteady compressible Euler equations with hightemperature gas effects. Solutions of some benchmark test cases are reported, and comparison between computational results of chemically equilibrium and perfect air flowfields is performed. The results of numerical simulation of onedimensional and twodimensional under and overexpanded nozzle flows with a moving region of energy supply are presented. Output nozzle parameters are calculated as functions of a number and time of burning of plasmatron arcs. The results obtained show a qualitative pattern of gas dynamics and thermal processes in the nozzle with unsteady energy supply demonstrating the displacement of the nozzle shock wave towards the nozzle outlet in the overexpanded nozzle flow in comparison to perfect gas flow.
 Abstract: Publication date: Available online 24 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): N.A. Brykov, V.N. Emelyanov, A.G. Karpenko, K.N. VolkovAbstractWhen gas flows at a high speed in a channel with a variable cross sectional area and highintensity energy supply, it experiences complicated physical and chemical processes producing hightemperature gas effects. Hightemperature gas effects are a key issue related to design and optimization of nozzles of plasmatron of alternating current. The finite volume method is applied to solve unsteady compressible Euler equations with hightemperature gas effects. Solutions of some benchmark test cases are reported, and comparison between computational results of chemically equilibrium and perfect air flowfields is performed. The results of numerical simulation of onedimensional and twodimensional under and overexpanded nozzle flows with a moving region of energy supply are presented. Output nozzle parameters are calculated as functions of a number and time of burning of plasmatron arcs. The results obtained show a qualitative pattern of gas dynamics and thermal processes in the nozzle with unsteady energy supply demonstrating the displacement of the nozzle shock wave towards the nozzle outlet in the overexpanded nozzle flow in comparison to perfect gas flow.
 Enriched two dimensional mixed finite element models for linear elasticity
with weak stress symmetry Abstract: Publication date: Available online 24 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Philippe R.B. Devloo, Sônia M. Gomes, Thiago O. Quinelato, Shudan TianAbstractThe purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry. These approximations are based on the application of enriched versions of classic Poissoncompatible spaces, for stress and displacement variables, and/or on enriched Stokescompatible space configurations, for the choice of rotation spaces used to weakly enforce stress symmetry. Accordingly, the stress space has to be adapted to ensure stability. Such enrichment procedures are done via space increments with extra bubble functions, which have their support on a single element (in the case of H1conforming approximations) or with vanishing normal components over element edges (in the case of H(div)conforming spaces). The advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed, in a way that the number of equations to be solved and the matrix structure are not affected. Enhanced approximations are observed when using the resulting enriched space configurations, which may have different orders of accuracy for the different variables. A general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables, individually. The use of enriched Poisson spaces improves the rates of convergence of stress divergence and displacement variables. Stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order, reaching the maximum rate given by the normal traces (which are not affected).
 Abstract: Publication date: Available online 24 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Philippe R.B. Devloo, Sônia M. Gomes, Thiago O. Quinelato, Shudan TianAbstractThe purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry. These approximations are based on the application of enriched versions of classic Poissoncompatible spaces, for stress and displacement variables, and/or on enriched Stokescompatible space configurations, for the choice of rotation spaces used to weakly enforce stress symmetry. Accordingly, the stress space has to be adapted to ensure stability. Such enrichment procedures are done via space increments with extra bubble functions, which have their support on a single element (in the case of H1conforming approximations) or with vanishing normal components over element edges (in the case of H(div)conforming spaces). The advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed, in a way that the number of equations to be solved and the matrix structure are not affected. Enhanced approximations are observed when using the resulting enriched space configurations, which may have different orders of accuracy for the different variables. A general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables, individually. The use of enriched Poisson spaces improves the rates of convergence of stress divergence and displacement variables. Stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order, reaching the maximum rate given by the normal traces (which are not affected).
 Numerical simulation of nanofluid convective heat transfer in an oblique
cavity with conductive edges equipped with a constant temperature heat
source: Entropy production analysis Abstract: Publication date: Available online 24 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Zhe Tian, Amin Shahsavar, Abdullah A.A.A. AlRashed, Sara RostamiAbstractIn the present work, the entropy production of Al2O3water nanofluid in an oblique cavity enclosure is examined. The side edges of the enclosure are at lower temperature and the top and down ones are insulated. Two thick conductive walls are placed on the low temperature edges. Furthermore, the center of the enclosure is equipped with a constant high temperature heat source. The enclosure is exposed to an inclined magnetic field. The governing nonlinear partial differential equations are continuity, Navier–Stokes and energy equations. These equations are solved using an opensource CFD software package (OpenFOAM). The influence of effective parameters includes Ra number, Hartman number, the angles of magnetic field and enclosure, nanoadditives concentration, and aspect ratio on the entropy production and the Bejan (Be) number are investigated. The results show that the minimum entropy production happened at low power magnetic fields. With the increase of the slope of the cavity, the entropy production rises. The addition of nanoadditives leads to an intensification in the entropy production and a reduction in the Be number.
 Abstract: Publication date: Available online 24 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Zhe Tian, Amin Shahsavar, Abdullah A.A.A. AlRashed, Sara RostamiAbstractIn the present work, the entropy production of Al2O3water nanofluid in an oblique cavity enclosure is examined. The side edges of the enclosure are at lower temperature and the top and down ones are insulated. Two thick conductive walls are placed on the low temperature edges. Furthermore, the center of the enclosure is equipped with a constant high temperature heat source. The enclosure is exposed to an inclined magnetic field. The governing nonlinear partial differential equations are continuity, Navier–Stokes and energy equations. These equations are solved using an opensource CFD software package (OpenFOAM). The influence of effective parameters includes Ra number, Hartman number, the angles of magnetic field and enclosure, nanoadditives concentration, and aspect ratio on the entropy production and the Bejan (Be) number are investigated. The results show that the minimum entropy production happened at low power magnetic fields. With the increase of the slope of the cavity, the entropy production rises. The addition of nanoadditives leads to an intensification in the entropy production and a reduction in the Be number.
 RANS simulations of terraindisrupted turbulent airflow at Hong Kong
International Airport Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Louis K.S. Tse, Yu Guan, Larry K.B. LiAbstractDuring a tropical cyclone, windshear can occur at Hong Kong International Airport (HKIA) owing to interactions between strong winds and the complex topography of the neighboring Lantau Island. For aviation safety, it is crucial to understand how such interactions arise under realistic but controlled meteorological conditions. In this study, we numerically simulate the turbulent airflow over Lantau Island and HKIA using OpenFOAM, an opensource platform for computational fluid dynamics. We use the platform to solve the Reynoldsaveraged Navier–Stokes equations via the finite volume method of discretization. We impose a neutrally stratified atmospheric boundary layer as the upwind condition, which is initialized with a logarithmic velocity profile in three different wind directions: southerly, southeasterly and easterly. For all three directions, we find multiple highspeed ∨shaped jets emanating from the mountain gaps of Lantau Island, giving rise to headwind and crosswind variations along the glide paths of HKIA. However, we find that it is primarily the southerly and southeasterly winds that are the most conducive to windshear. For these two wind directions, we find that windshear is most likely to occur along the glide paths of the two existing runways because these are the closest to Lantau Island itself. The third runway, which is currently under construction, is the least likely to suffer from windshear. By showing how airflow disturbances arise from the complex topography of Lantau Island, this study contributes to safer and more efficient flight operations at HKIA.
 Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Louis K.S. Tse, Yu Guan, Larry K.B. LiAbstractDuring a tropical cyclone, windshear can occur at Hong Kong International Airport (HKIA) owing to interactions between strong winds and the complex topography of the neighboring Lantau Island. For aviation safety, it is crucial to understand how such interactions arise under realistic but controlled meteorological conditions. In this study, we numerically simulate the turbulent airflow over Lantau Island and HKIA using OpenFOAM, an opensource platform for computational fluid dynamics. We use the platform to solve the Reynoldsaveraged Navier–Stokes equations via the finite volume method of discretization. We impose a neutrally stratified atmospheric boundary layer as the upwind condition, which is initialized with a logarithmic velocity profile in three different wind directions: southerly, southeasterly and easterly. For all three directions, we find multiple highspeed ∨shaped jets emanating from the mountain gaps of Lantau Island, giving rise to headwind and crosswind variations along the glide paths of HKIA. However, we find that it is primarily the southerly and southeasterly winds that are the most conducive to windshear. For these two wind directions, we find that windshear is most likely to occur along the glide paths of the two existing runways because these are the closest to Lantau Island itself. The third runway, which is currently under construction, is the least likely to suffer from windshear. By showing how airflow disturbances arise from the complex topography of Lantau Island, this study contributes to safer and more efficient flight operations at HKIA.
 A twogrid MMOC finite element method for nonlinear variableorder
timefractional mobile/immobile advection–diffusion equations Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Chuanjun Chen, Huan Liu, Xiangcheng Zheng, Hong WangAbstractA fully discrete twogrid modified method of characteristics (MMOC) scheme is proposed for nonlinear variableorder timefractional advection–diffusion equations in two space dimensions. The MMOC is used to handle the advectiondominated transport and the twogrid method is designed for efficiently solving the resulting nonlinear system. Optimal L2 error estimates are derived for both the MMOC scheme and the corresponding twogrid MMOC scheme. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method.
 Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Chuanjun Chen, Huan Liu, Xiangcheng Zheng, Hong WangAbstractA fully discrete twogrid modified method of characteristics (MMOC) scheme is proposed for nonlinear variableorder timefractional advection–diffusion equations in two space dimensions. The MMOC is used to handle the advectiondominated transport and the twogrid method is designed for efficiently solving the resulting nonlinear system. Optimal L2 error estimates are derived for both the MMOC scheme and the corresponding twogrid MMOC scheme. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method.
 A Hybrid HighOrder method for the incompressible Navier–Stokes problem
robust for large irrotational body forces Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Daniel Castanon Quiroz, Daniele A. Di PietroAbstractWe develop a novel Hybrid HighOrder method for the incompressible Navier–Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective contributions in the momentum equation formulated in terms of a globally divergencefree velocity reconstruction. Two key properties are mimicked at the discrete level, namely the invariance of the velocity with respect to irrotational body forces and the nondissipativity of the convective term. A full convergence analysis is carried out, showing optimal orders of convergence under a smallness condition involving only the solenoidal part of the body force. The performance of the method is illustrated by a complete panel of numerical tests, including comparisons that highlight the benefits with respect to more standard formulations.
 Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Daniel Castanon Quiroz, Daniele A. Di PietroAbstractWe develop a novel Hybrid HighOrder method for the incompressible Navier–Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective contributions in the momentum equation formulated in terms of a globally divergencefree velocity reconstruction. Two key properties are mimicked at the discrete level, namely the invariance of the velocity with respect to irrotational body forces and the nondissipativity of the convective term. A full convergence analysis is carried out, showing optimal orders of convergence under a smallness condition involving only the solenoidal part of the body force. The performance of the method is illustrated by a complete panel of numerical tests, including comparisons that highlight the benefits with respect to more standard formulations.
 A self adjusting multirate algorithm for robust time discretization of
partial differential equations Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): L. Bonaventura, F. Casella, L. Delpopolo Carciopolo, A. RanadeAbstractWe show the distinctive potential advantages of a self adjusting multirate method based on diagonally implicit solvers for the robust time discretization of partial differential equations. The properties of the specific ODE methods considered are reviewed, with special focus on the TRBDF2 solver. A general expression for the stability function of a generic one stage multirate method is derived, which allows to study numerically the stability properties of the proposed algorithm in a number of examples relevant for applications. Several numerical experiments, aimed at the time discretization of hyperbolic partial differential equations, demonstrate the efficiency and accuracy of the resulting approach.
 Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): L. Bonaventura, F. Casella, L. Delpopolo Carciopolo, A. RanadeAbstractWe show the distinctive potential advantages of a self adjusting multirate method based on diagonally implicit solvers for the robust time discretization of partial differential equations. The properties of the specific ODE methods considered are reviewed, with special focus on the TRBDF2 solver. A general expression for the stability function of a generic one stage multirate method is derived, which allows to study numerically the stability properties of the proposed algorithm in a number of examples relevant for applications. Several numerical experiments, aimed at the time discretization of hyperbolic partial differential equations, demonstrate the efficiency and accuracy of the resulting approach.
 Error analysis of a finite element method with GMMP temporal
discretisation for a timefractional diffusion equation Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Chaobao Huang, Martin StynesAbstractA timefractional initial–boundary value problem is considered, where the spatial domain has dimension d∈{1,2,3}, the spatial differential operator is a standard elliptic operator, and the time derivative is a Caputo derivative of order α∈(0,1). To discretise in space we use a standard piecewisepolynomial finite element method, while for the temporal discretisation the GMMP scheme (a variant of the GrünwaldLetnikov scheme) is used on a uniform mesh. The analysis of the GMMP scheme for solutions that exhibit a typical weak singularity at the initial time t=0 has not previously been considered in the literature. A global convergence result is derived in L∞(L2), then a more delicate analysis of the error in this norm shows that, away from t=0, the method attains optimalrate convergence. Numerical results confirm the sharpness of the theoretical error bounds.
 Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Chaobao Huang, Martin StynesAbstractA timefractional initial–boundary value problem is considered, where the spatial domain has dimension d∈{1,2,3}, the spatial differential operator is a standard elliptic operator, and the time derivative is a Caputo derivative of order α∈(0,1). To discretise in space we use a standard piecewisepolynomial finite element method, while for the temporal discretisation the GMMP scheme (a variant of the GrünwaldLetnikov scheme) is used on a uniform mesh. The analysis of the GMMP scheme for solutions that exhibit a typical weak singularity at the initial time t=0 has not previously been considered in the literature. A global convergence result is derived in L∞(L2), then a more delicate analysis of the error in this norm shows that, away from t=0, the method attains optimalrate convergence. Numerical results confirm the sharpness of the theoretical error bounds.
 An anisotropic PDE model for image inpainting
 Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Abdul Halim, B.V. Rathish KumarAbstractA PDE model based on an image specific auto generated multiwell potential function and a 4th order anisotropic diffusion with good performance with respect to loss due to smoothing effects has been proposed for grayscale image inpainting. An unconditionally stable numerical scheme using the notion of convexity splitting in time and Fourier spectral method in space has been derived. The stable scheme is both consistent and convergent. Numerical computations are done for some standard test images and results are compared with the results of other existing models in the literature using image analysis specs such as PSNR, SNR, and SSIM.
 Abstract: Publication date: Available online 23 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Abdul Halim, B.V. Rathish KumarAbstractA PDE model based on an image specific auto generated multiwell potential function and a 4th order anisotropic diffusion with good performance with respect to loss due to smoothing effects has been proposed for grayscale image inpainting. An unconditionally stable numerical scheme using the notion of convexity splitting in time and Fourier spectral method in space has been derived. The stable scheme is both consistent and convergent. Numerical computations are done for some standard test images and results are compared with the results of other existing models in the literature using image analysis specs such as PSNR, SNR, and SSIM.
 An oil–water twophase reservoir numerical simulation coupled with
dynamic capillary force based on the fullimplicit method Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Shuoliang Wang, Chunlei Yu, Guoqiang Sang, Rongze Yu, Feng ChengAbstractCapillary pressure is a key parameter in reservoir numerical simulation. It is generally considered that the capillary force is an only function of water saturation in the current traditional reservoir numerical simulation methods. However, large amount of indoor experiments have shown that capillary force is not only dependent on the water saturation, but also the velocity of fluid flow. In this work, we introduce the dynamic capillary pressure to the oil–water twophase flow equations. Then, we discrete the modified oil and water phase basic governing equation with difference methods. The full implicit method is selected to solve the pressure equation. Finally, a numerical simulation method coupled with the dynamic capillary force based on the fullimplicit method is established. We discussed the effect of dynamic capillary pressure on the oil–water twophase flow in twodimensional homogeneous and heterogeneous porous media. Comparing to static capillary pressure model, the dynamic capillary pressure model predicts higher watersaturation near the injection and production wells. In addition, a “funnel” like pressure profile appears in the water saturation profile. At low water cut stage, the rate of water cut increase is relatively high, while opposite is true at high water cut stage. We also calibrated our newly proposed model by comparing to the analytical method and the actual oil field production data. Our work should provide important insights into the effect of dynamic capillary force on the oil–water twophase flow and better estimation of flow behavior in oil and gas reservoir.
 Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Shuoliang Wang, Chunlei Yu, Guoqiang Sang, Rongze Yu, Feng ChengAbstractCapillary pressure is a key parameter in reservoir numerical simulation. It is generally considered that the capillary force is an only function of water saturation in the current traditional reservoir numerical simulation methods. However, large amount of indoor experiments have shown that capillary force is not only dependent on the water saturation, but also the velocity of fluid flow. In this work, we introduce the dynamic capillary pressure to the oil–water twophase flow equations. Then, we discrete the modified oil and water phase basic governing equation with difference methods. The full implicit method is selected to solve the pressure equation. Finally, a numerical simulation method coupled with the dynamic capillary force based on the fullimplicit method is established. We discussed the effect of dynamic capillary pressure on the oil–water twophase flow in twodimensional homogeneous and heterogeneous porous media. Comparing to static capillary pressure model, the dynamic capillary pressure model predicts higher watersaturation near the injection and production wells. In addition, a “funnel” like pressure profile appears in the water saturation profile. At low water cut stage, the rate of water cut increase is relatively high, while opposite is true at high water cut stage. We also calibrated our newly proposed model by comparing to the analytical method and the actual oil field production data. Our work should provide important insights into the effect of dynamic capillary force on the oil–water twophase flow and better estimation of flow behavior in oil and gas reservoir.
 A weakform RBFgenerated finite difference method
 Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Mozhgan Jabalameli, Davoud MirzaeiAbstractIn this paper, the idea of direct discretization via radial basis functions (RBFs) is applied on a local Petrov–Galerkin test space of a partial differential equation (PDE). This results to a weakbased RBFgenerated finite difference (RBF–FD) scheme that possesses some useful properties. The error and stability issues are considered. When the PDE solution or the basis function has low smoothness, the new method gives more accurate results than the already wellestablished strongbased collocation methods. Although the method uses a Galerkin formulation, it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over nonconnected, independent and wellshaped subdomains. Some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method.
 Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Mozhgan Jabalameli, Davoud MirzaeiAbstractIn this paper, the idea of direct discretization via radial basis functions (RBFs) is applied on a local Petrov–Galerkin test space of a partial differential equation (PDE). This results to a weakbased RBFgenerated finite difference (RBF–FD) scheme that possesses some useful properties. The error and stability issues are considered. When the PDE solution or the basis function has low smoothness, the new method gives more accurate results than the already wellestablished strongbased collocation methods. Although the method uses a Galerkin formulation, it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over nonconnected, independent and wellshaped subdomains. Some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method.
 A highorder threescale approach for predicting thermomechanical
properties of porous materials with interior surface radiation Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Zhiqiang Yang, Yi Sun, Tianyu Guan, Hao DongAbstractA highorder threescale approach developed in this work to analyze thermomechanical properties of porous materials with interior surface radiation is systematically studied. The microstructures of the porous structures are described by periodical layout of local cells on the microscopic domain and mesoscopic domain, and surface radiation effect at microscale and mesoscale is also investigated. At first, the threescale formulas based on reiterated homogenization and highorder asymptotic expansion are established, and the local cell solutions in microscale and mesoscale are also defined. Then, two kinds of homogenized parameters are evaluated by upscaling methods, and the homogenized equations are derived on the whole structure. Further, heat flux and strain fields are constructed as the threescale asymptotic solutions by assembling the higherorder unit cell solutions and homogenized solutions. The significant features of the proposed approach are an asymptotic highorder homogenization that does not require higher order continuity of the macroscale solutions and a new highorder threescale formula derived for analyzing the coupled problems. Finally, some representative examples are proposed to verify the presented methods. They show that the threescale asymptotic expansions introduced in this paper are efficient and valid for predicting the thermomechanical properties of the porous materials with multiple spatial scales.
 Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Zhiqiang Yang, Yi Sun, Tianyu Guan, Hao DongAbstractA highorder threescale approach developed in this work to analyze thermomechanical properties of porous materials with interior surface radiation is systematically studied. The microstructures of the porous structures are described by periodical layout of local cells on the microscopic domain and mesoscopic domain, and surface radiation effect at microscale and mesoscale is also investigated. At first, the threescale formulas based on reiterated homogenization and highorder asymptotic expansion are established, and the local cell solutions in microscale and mesoscale are also defined. Then, two kinds of homogenized parameters are evaluated by upscaling methods, and the homogenized equations are derived on the whole structure. Further, heat flux and strain fields are constructed as the threescale asymptotic solutions by assembling the higherorder unit cell solutions and homogenized solutions. The significant features of the proposed approach are an asymptotic highorder homogenization that does not require higher order continuity of the macroscale solutions and a new highorder threescale formula derived for analyzing the coupled problems. Finally, some representative examples are proposed to verify the presented methods. They show that the threescale asymptotic expansions introduced in this paper are efficient and valid for predicting the thermomechanical properties of the porous materials with multiple spatial scales.
 European option pricing under stochastic volatility jumpdiffusion models
with transaction cost Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yingxu Tian, Haoyan ZhangAbstractIn this paper, we consider an underlying general stochastic volatility jumpdiffusion model. Option pricing under this general model with transaction costs will lead to handling with nonlinear partial integrodifferential equations (here after PIDE). In this case, option replication in a discretetime framework with transaction costs and the nonuniqueness option pricing in such incomplete market will be studied. Observing and introducing a traded proxy for the volatility in the modern market, we acquire a nonlinear PIDE in the advent of transaction costs. Under appropriate regularity conditions, the existence of the strong solution to this pricing problem has been proved. The corresponding selffinancing and positions readjustment for pricing a portfolio will also be discussed in the end.
 Abstract: Publication date: Available online 20 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yingxu Tian, Haoyan ZhangAbstractIn this paper, we consider an underlying general stochastic volatility jumpdiffusion model. Option pricing under this general model with transaction costs will lead to handling with nonlinear partial integrodifferential equations (here after PIDE). In this case, option replication in a discretetime framework with transaction costs and the nonuniqueness option pricing in such incomplete market will be studied. Observing and introducing a traded proxy for the volatility in the modern market, we acquire a nonlinear PIDE in the advent of transaction costs. Under appropriate regularity conditions, the existence of the strong solution to this pricing problem has been proved. The corresponding selffinancing and positions readjustment for pricing a portfolio will also be discussed in the end.
 Uncertainty propagation using WienerLinear Bspline wavelet expansion
 Abstract: Publication date: Available online 19 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Navjot Kaur, Kavita GoyalAbstractIn this paper, we have constructed a scheme combining generalized Polynomial Chaos (gPC) representation and Bspline wavelets. To begin with, semiorthogonal compactly supported Bspline wavelets are constructed for the bounded interval [0,1] which are used for PC expansion of possible stochastic processes. To compute the deterministic coefficients of expansion, we have applied Galerkin projection on uncertain data and the solution variables. Then, to ascertain the behavior of the random process, the system of equations obtained from projection are integrated using fourth order Runge–Kutta method. To handle the nonlinearity, we have compared Galerkin projection with pseudospectral projection. The procedure is illustrated through three model problems of real life importance. We conclude that Galerkin approximation performs better in comparison to pseudospectral approach which is numerically expected. Also, it has been observed that the wavelet function based expansion shows superior results as compared to scaling function based expansion.
 Abstract: Publication date: Available online 19 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Navjot Kaur, Kavita GoyalAbstractIn this paper, we have constructed a scheme combining generalized Polynomial Chaos (gPC) representation and Bspline wavelets. To begin with, semiorthogonal compactly supported Bspline wavelets are constructed for the bounded interval [0,1] which are used for PC expansion of possible stochastic processes. To compute the deterministic coefficients of expansion, we have applied Galerkin projection on uncertain data and the solution variables. Then, to ascertain the behavior of the random process, the system of equations obtained from projection are integrated using fourth order Runge–Kutta method. To handle the nonlinearity, we have compared Galerkin projection with pseudospectral projection. The procedure is illustrated through three model problems of real life importance. We conclude that Galerkin approximation performs better in comparison to pseudospectral approach which is numerically expected. Also, it has been observed that the wavelet function based expansion shows superior results as compared to scaling function based expansion.
 Parameter identification for open cell aluminium foams using inverse
calculation Abstract: Publication date: Available online 16 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): T. Bleistein, S. Diebels, A. JungAbstractOpen cell metal foams are interesting lightweight materials, which are used in a variety of applications. Effected by the complex microstructure of open cell metal foams the macroscopic material behaviour cannot be described with a von Mises yield criterion. In comparison to bulk metals, open cell metal foams collapse under hydrostatic compression and tension. Modelling of this behaviour is realised by using a closed yield surface to describe the plastic behaviour. This contribution considers the asymmetric yield surfaces measured for open cell metal foams with 10 and 20 ppi. A Finite Element implementation of a single closed yield surface is introduced. Furthermore, a new approach to identify the material parameters of the yield surface as well as the structural Young’s and hardening moduli is introduced. The simulations are realised with a Finite Element Method and the numerical results are compared with the experiments. The numerical results are fitted to the experimental data by modifying the material parameters of the model in an iterative optimisation process. The optimised material parameters are used to describe different loading conditions for open cell aluminium foams. The results of the simulation are compared with the multiaxial experiments for the sake of validation.
 Abstract: Publication date: Available online 16 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): T. Bleistein, S. Diebels, A. JungAbstractOpen cell metal foams are interesting lightweight materials, which are used in a variety of applications. Effected by the complex microstructure of open cell metal foams the macroscopic material behaviour cannot be described with a von Mises yield criterion. In comparison to bulk metals, open cell metal foams collapse under hydrostatic compression and tension. Modelling of this behaviour is realised by using a closed yield surface to describe the plastic behaviour. This contribution considers the asymmetric yield surfaces measured for open cell metal foams with 10 and 20 ppi. A Finite Element implementation of a single closed yield surface is introduced. Furthermore, a new approach to identify the material parameters of the yield surface as well as the structural Young’s and hardening moduli is introduced. The simulations are realised with a Finite Element Method and the numerical results are compared with the experiments. The numerical results are fitted to the experimental data by modifying the material parameters of the model in an iterative optimisation process. The optimised material parameters are used to describe different loading conditions for open cell aluminium foams. The results of the simulation are compared with the multiaxial experiments for the sake of validation.
 Revisiting performance of BiCGStab methods for solving systems with
multiple righthand sides Abstract: Publication date: Available online 14 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): B. KrasnopolskyAbstractThe paper discusses the efficiency of the classical BiCGStab method and several of its modifications for solving systems with multiple righthand side vectors. These iterative methods are widely used for solving systems with large sparse matrices. The paper presents execution time analytical model for the time to solve the systems. The BiCGStab method and several modifications including the Reordered BiCGStab and Pipelined BiCGStab methods are analysed and the range of applicability for each method providing the best execution time is highlighted. The results of the analytical model are validated by the numerical experiments and compared with results of other authors. The presented results demonstrate an increasing role of the vector operations when performing simulations with multiple righthand side vectors. The proposed merging of vector operations allows to reduce the memory traffic and improve performance of the calculations by about 30%.
 Abstract: Publication date: Available online 14 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): B. KrasnopolskyAbstractThe paper discusses the efficiency of the classical BiCGStab method and several of its modifications for solving systems with multiple righthand side vectors. These iterative methods are widely used for solving systems with large sparse matrices. The paper presents execution time analytical model for the time to solve the systems. The BiCGStab method and several modifications including the Reordered BiCGStab and Pipelined BiCGStab methods are analysed and the range of applicability for each method providing the best execution time is highlighted. The results of the analytical model are validated by the numerical experiments and compared with results of other authors. The presented results demonstrate an increasing role of the vector operations when performing simulations with multiple righthand side vectors. The proposed merging of vector operations allows to reduce the memory traffic and improve performance of the calculations by about 30%.
 Electricity generation in a microbial fuel cell with textile carbon fibre
anodes Abstract: Publication date: Available online 13 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Peter Farber, Jens Gräbel, Norman Kroppen, Liesa Pötschke, Dirk Roos, Miriam Rosenbaum, Georg Stegschuster, Peer UeberholzAbstractCommercial Computational Fluid Dynamics (CFD) codes offer a great flexibility to model complex 3D geometries. They have many physical models on board, nevertheless reactions in a Microbial Fuel Cell (MFC) are not included. In this paper, we discuss the extension of Ansys Fluent commercial CFD code to simulate a model of an anode in a Microbial Fuel Cell. The biofilm around the anode is a mixed culture dominated by Geobacter sulfurreducens and is treated as a conductive material. Besides the stationary 3D Navier–Stokes equation for fluid flow and the species balance equation for acetate in the water and in the biofilm, the model includes a model for the species mass fraction of acetate at the boundary between water and biofilm. Furthermore, we added a sink for acetate as well as a source for electrons and a stationary electric potential equation in the biofilm. Using this extended commercial CFD code and a 128 core compute cluster allowed us to explore the impact of different textile carbon fibre based anode configurations on the electrical performance of the anode in a MFC. The results show that the size of the outer surface of the biofilm determines the quantity of the electrical power delivered by the biofilm.
 Abstract: Publication date: Available online 13 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Peter Farber, Jens Gräbel, Norman Kroppen, Liesa Pötschke, Dirk Roos, Miriam Rosenbaum, Georg Stegschuster, Peer UeberholzAbstractCommercial Computational Fluid Dynamics (CFD) codes offer a great flexibility to model complex 3D geometries. They have many physical models on board, nevertheless reactions in a Microbial Fuel Cell (MFC) are not included. In this paper, we discuss the extension of Ansys Fluent commercial CFD code to simulate a model of an anode in a Microbial Fuel Cell. The biofilm around the anode is a mixed culture dominated by Geobacter sulfurreducens and is treated as a conductive material. Besides the stationary 3D Navier–Stokes equation for fluid flow and the species balance equation for acetate in the water and in the biofilm, the model includes a model for the species mass fraction of acetate at the boundary between water and biofilm. Furthermore, we added a sink for acetate as well as a source for electrons and a stationary electric potential equation in the biofilm. Using this extended commercial CFD code and a 128 core compute cluster allowed us to explore the impact of different textile carbon fibre based anode configurations on the electrical performance of the anode in a MFC. The results show that the size of the outer surface of the biofilm determines the quantity of the electrical power delivered by the biofilm.
 A block triplerelaxationtime lattice Boltzmann model for nonlinear
anisotropic convection–diffusion equations Abstract: Publication date: Available online 9 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yong Zhao, Yao Wu, Zhenhua Chai, Baochang ShiAbstractA block triplerelaxationtime (BTriRT) lattice Boltzmann model for general nonlinear anisotropic convection–diffusion equations (NACDEs) is proposed, and the Chapman–Enskog analysis shows that the present BTriRT model can recover the NACDEs correctly. There are some striking features of the present BTriRT model: firstly, the relaxation matrix of BTriRT model is partitioned into three relaxation parameter blocks, rather than a diagonal matrix in general multiplerelaxationtime (MRT) model; secondly, based on the analysis of halfway bounceback (HBB) scheme for Dirichlet boundary conditions, we obtain an expression to determine the relaxation parameters; thirdly, the anisotropic diffusion tensor can be recovered by the relaxation parameter block that corresponds to the firstorder moment of nonequilibrium distribution function. A number of simulations of isotropic and anisotropic convection–diffusion equations are conducted to validate the present BTriRT model. The results indicate that the present model has a secondorder accuracy in space, and is also more accurate and more stable than some available lattice Boltzmann models.
 Abstract: Publication date: Available online 9 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Yong Zhao, Yao Wu, Zhenhua Chai, Baochang ShiAbstractA block triplerelaxationtime (BTriRT) lattice Boltzmann model for general nonlinear anisotropic convection–diffusion equations (NACDEs) is proposed, and the Chapman–Enskog analysis shows that the present BTriRT model can recover the NACDEs correctly. There are some striking features of the present BTriRT model: firstly, the relaxation matrix of BTriRT model is partitioned into three relaxation parameter blocks, rather than a diagonal matrix in general multiplerelaxationtime (MRT) model; secondly, based on the analysis of halfway bounceback (HBB) scheme for Dirichlet boundary conditions, we obtain an expression to determine the relaxation parameters; thirdly, the anisotropic diffusion tensor can be recovered by the relaxation parameter block that corresponds to the firstorder moment of nonequilibrium distribution function. A number of simulations of isotropic and anisotropic convection–diffusion equations are conducted to validate the present BTriRT model. The results indicate that the present model has a secondorder accuracy in space, and is also more accurate and more stable than some available lattice Boltzmann models.
 Local ultraconvergence of high order finite element method by
interpolation postprocessing technique for elliptic problems with constant
coefficients Abstract: Publication date: Available online 9 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Wenming He, Xiong Liu, Jin XiaoAbstractAssume that u(x) satisfies the problem Lu(x)≡−∂∂xi(aij∂u∂xj)=f(x),∀x∈Ω,u(x)=0,∀x∈∂Ω. In this article, using interpolation postprocessing technique, we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u(x) using piecewise polynomials of degrees bik(k≥3) over a rectangular partition. Assume that k≥3 is odd and x0 is an interior vertex satisfying ρ(x0,∂Ω)≥c. Using the new interpolation postprocessing formula presented in this study, we show that the primal variable and the derivative of the postprocessed finite element solution using piecewise of degrees bik(k≥3) at x0 converge to the primal variable and the derivative of the exact solution with order O(hk+3 lnh ) under suitable regularity and mesh conditions, respectively. Finally, we use numerical experiments to illustrate our theoretical findings.
 Abstract: Publication date: Available online 9 December 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Wenming He, Xiong Liu, Jin XiaoAbstractAssume that u(x) satisfies the problem Lu(x)≡−∂∂xi(aij∂u∂xj)=f(x),∀x∈Ω,u(x)=0,∀x∈∂Ω. In this article, using interpolation postprocessing technique, we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u(x) using piecewise polynomials of degrees bik(k≥3) over a rectangular partition. Assume that k≥3 is odd and x0 is an interior vertex satisfying ρ(x0,∂Ω)≥c. Using the new interpolation postprocessing formula presented in this study, we show that the primal variable and the derivative of the postprocessed finite element solution using piecewise of degrees bik(k≥3) at x0 converge to the primal variable and the derivative of the exact solution with order O(hk+3 lnh ) under suitable regularity and mesh conditions, respectively. Finally, we use numerical experiments to illustrate our theoretical findings.
 A segregated spectral finite element method for the 2D transient
incompressible Navier–Stokes equations Abstract: Publication date: Available online 29 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Wenqiang He, Guoliang Qin, Jingxiang Lin, Cheng JiaAbstractIn this paper, the spectral element approximation and the velocity–pressuredecoupling method implementing the SIMPLE (semiimplicit method for pressure linked equations) algorithm are first combined to form a highorder segregated scheme for the solution of the twodimensional transient incompressible Navier–Stokes equations. In contrast to previous segregated finite element methods based on the SIMPLE algorithm, the pressure equation is derived from the continuity equation using the element matrices to ensure convergence. Highorder element basis functions are adopted, which greatly reduces the number of nodal points used in the calculation. The validation test that has an analytical solution demonstrates the high accuracy and convergence rate of the method. The flow in a liddriven cavity with different inclination angles and the flow over a backwardfacing step are investigated to further illustrate the performance of the scheme. The computed results are in excellent agreement with the benchmark solutions. The almost periodic solution for the flow of Re=10000 in a liddriven square cavity is also captured by the present scheme.
 Abstract: Publication date: Available online 29 November 2019Source: Computers & Mathematics with ApplicationsAuthor(s): Wenqiang He, Guoliang Qin, Jingxiang Lin, Cheng JiaAbstractIn this paper, the spectral element approximation and the velocity–pressuredecoupling method implementing the SIMPLE (semiimplicit method for pressure linked equations) algorithm are first combined to form a highorder segregated scheme for the solution of the twodimensional transient incompressible Navier–Stokes equations. In contrast to previous segregated finite element methods based on the SIMPLE algorithm, the pressure equation is derived from the continuity equation using the element matrices to ensure convergence. Highorder element basis functions are adopted, which greatly reduces the number of nodal points used in the calculation. The validation test that has an analytical solution demonstrates the high accuracy and convergence rate of the method. The flow in a liddriven cavity with different inclination angles and the flow over a backwardfacing step are investigated to further illustrate the performance of the scheme. The computed results are in excellent agreement with the benchmark solutions. The almost periodic solution for the flow of Re=10000 in a liddriven square cavity is also captured by the present scheme.