Abstract: Publication date: Available online 10 March 2017 Source:Computers & Mathematics with Applications Author(s): Rui M.P. Almeida, Stanislav N. Antontsev, José C.M. Duque The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of degree k ≥ 1 . Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.

Abstract: Publication date: Available online 9 March 2017 Source:Computers & Mathematics with Applications Author(s): Gheorghe Moroşanu, Mihai Nechita We are concerned with Hanusse-type chemical models with diffusions. We show that some bounded invariant sets ⊂ R N found for the ODE Hanusse-type models (corresponding to the case when diffusions are neglected) can be used to define invariant sets ⊂ L ∞ ( Ω ) N with respect to the corresponding Hanusse-type PDE models (involving diffusions), where Ω ⊂ R n , n ≤ 3 , denotes the reaction domain. Simulations for both the ODE and PDE Hanusse-type models are performed for suitable coefficients of the polynomials representing the reaction terms, showing that the attractors for the ODE model are also attractors, in fact the only attractors, for the PDE model.

Abstract: Publication date: Available online 9 March 2017 Source:Computers & Mathematics with Applications Author(s): Jichun Li, Cengke Shi, Chi-Wang Shu Simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain Maxwell’s equations than the standard Maxwell’s equations in free space. In this paper, we develop and analyze a non-dissipative discontinuous Galerkin (DG) method for solving the Maxwell’s equations in Drude metamaterials. Previous discontinuous Galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non-dissipative but sub-optimal, or dissipative and optimal. Our method uses a different and simple choice of numerical fluxes, achieving provable non-dissipative stability and optimal error estimates simultaneously. We prove the stability and optimal error estimates for both semi- and fully discrete DG schemes, with the leap-frog time discretization for the fully discrete case. Numerical results are given to demonstrate that the DG method can solve metamaterial Maxwell’s equations effectively.

Abstract: Publication date: Available online 9 March 2017 Source:Computers & Mathematics with Applications Author(s): Zhihui Zhao, Hong Li, Zhendong Luo The convergence of space–time continuous Galerkin (STCG) method for the Sobolev equations with convection-dominated terms is studied in this article. It allows variable time steps and the change of the spatial mesh from one time interval to the next, which can make this method suitable for numerical simulations on unstructured grids. We prove the existence and uniqueness of the approximate solution and get the optimal convergence rates in L ∞ ( H 1 ) norm which do not require any restriction assumptions on the space and time mesh size. Finally, some numerical examples are designed to validate the high efficiency of the method showed herein and to confirm the correctness of the theoretical analysis.

Abstract: Publication date: Available online 8 March 2017 Source:Computers & Mathematics with Applications Author(s): Yibao Li, Yongho Choi, Junseok Kim In this work, we propose a fast and efficient adaptive time step procedure for the Cahn–Hilliard equation. The temporal evolution of the Cahn–Hilliard equation has multiple time scales. For spinodal decomposition simulation, an initial random perturbation evolves on a fast time scale, and later coarsening evolves on a very slow time scale. Therefore, if a small time step is used to capture the fast dynamics, the computation is quite costly. On the other hand, if a large time step is used, fast time evolutions may be missed. Hence, it is essential to use an adaptive time step method to simulate phenomena with multiple time scales. The proposed time adaptivity algorithm is based on the discrete maximum norm of the difference between two consecutive time step numerical solutions. Numerical experiments in one, two, and three dimensions are presented to demonstrate the performance and effectiveness of the adaptive time-stepping algorithm.

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Cheng-Cheng Zhu, Wan-Tong Li, Fei-Ying Yang This paper is concerned with traveling wave solutions of a nonlocal dispersal Susceptible–Infective–Removal–Healing (for short SIRH ) model with relapse. It is found that the existence and nonexistence of traveling waves of the system are not only determined by the critical wave speed c ∗ , but also by the basic reproduction number R 0 of the corresponding system of ordinary differential equations. More precisely, we use Schauder’s fixed-point theorem to obtain the existence of traveling waves for R 0 > 1 and c > c ∗ , and the nonexistence of traveling waves for R 0 > 1 and 0 < c < c ∗ . Some numerical simulations and discussions are also provided to illustrate our analytical results.

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Wei-Ru Xu, Guo-Liang Chen Let R ∈ C m × m and S ∈ C n × n be nontrivial k -involutions if their minimal polynomials are both x k − 1 for some k ≥ 2 , i.e., R k − 1 = R − 1 ≠ ± I and S k − 1 = S − 1 ≠ ± I . We say that A ∈ C m × n is ( R , S , μ ) -symmetric if R A S − 1 = ζ μ A , and A is ( R , S , α , μ ) -symmetric if R A S − α = ζ μ A with α , μ ∈ { 0 , 1 , … , k − 1 } and α ≠ 0 . Let S be one of the subsets of all ( R , S , μ ) -symmetric and ( R , S , α , μ ) -symmetric matrices. Given X ∈ C n × r , Y ∈ C s × m , B ∈ C m × r and D ∈ C s × n , we characterize the matrices A in S that minimize ‖ A X − B ‖ 2 + ‖ Y A − D ‖ 2 (Frobenius norm) under the assumption that R and S are unitary. Moreover, among the set S ( X , Y , B , D ) PubDate: 2017-03-08T12:58:43Z

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): N. Noormohammadi, B. Boroomand In this paper a novel method is presented to construct singular basis functions for solving harmonic and bi-harmonic problems with weak singularities. Such bases are found without any knowledge of the singularity order. The singular bases are constructed by choosing a series as a tensor product of Chebyshev polynomials and trigonometrical functions, in radial and angular directions respectively, and applying weak form of the governing equation. With such features, the singular bases are categorized as the equilibrated basis functions. The constructed singular functions can be utilized as a complementary part to the smooth part of the approximation in the solution of problems with singularities. To demonstrate the efficiency of employing such singular bases, they are used in a boundary node method. Through the solution of some examples, selected from the well-known literature, the capability of the method is shown. It will be demonstrated that the main function and its derivatives are excellently approximated at very close neighborhood of the singular point. The method may especially be found useful for those who research on the eXtended Finite Element Method or similar ideas.

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Azael Capetillo, Fernando Ibarra Advances in emission control technologies have seen the introduction of Selective Catalyst Reduction (SCR) systems as a method for NOx decontamination in light and heavy duty vehicles. SCR systems make use of a urea–water solution (UWS) injected directly into the exhaust gas stream for the reduction of NOx contaminants to Nitrogen (N2) over a monolith catalyst [2]. The effectiveness of an SCR system depends on many factors including the type of catalysts, the injection and mixing pattern of the UWS, temperature and more [1]. Spray analysis involves multi-phase flow phenomena and requires the numerical solution of the conservation and transport equations for the gas and the liquid phase simultaneously. Spray/wall interaction mechanisms such as droplet splash, spread, rebound or stick are complex to model and directly affected by the injection parameters [4]. The accurate modelling of the UWS injector can help in the prediction of phenomena such as wall film formation, droplet evaporation and urea crystallization [7]. This study presents a series of multi-phase numerical analyses, computed with the commercial software AVL Fire 2014 v, to measure the impact of injection velocity, spray angle and droplet size, in the overall performance of an SCR system. The analysis consisted of a completely mixed turbulent flow, solved using a two equations turbulence model ( k − z e t a − f ) . The interaction of the injected particles was solved with an Euler/Lagrange approach, the liquid phase calculation was based on the statistical Discrete Droplet Method interacting with the numerical solution of the conservation equations of the flow pattern. It was found that the injection parameters had an impact on the final results of the ammonia uniformity index (NH 3 UI) and wall film formation on the SCR system. The method applied in this work successfully predicted the performance of an SCR system. Moreover, a series of response surfaces were created based on a linear regression model which allowed for further design optimization outside of the initial experimental space.

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Štěpán Papáček, Benn Macdonald, Ctirad Matonoha Fluorescence recovery after photobleaching (FRAP) is a widely used method to analyze (usually using fluorescence microscopy) the mobility of either fluorescently tagged or autofluorescent (e.g., photosynthetic) proteins in living cells. The FRAP method resides in imaging the recovery of fluorescence intensity over time in a region of interest previously bleached by a high-intensity laser pulse. While the basic principles of FRAP are simple and the experimental setup is usually fixed, quantitative FRAP data analysis is not well developed. Different models and numerical procedures are used for the underlying model parameter estimation without knowledge of how robust the methods are, i.e., the parameter inference step is not currently well established. In this paper we rigorously formulate the inverse problem of model parameter estimation (including the sensitivity analysis), making possible the comparison of different FRAP parameter inference methods. Then, in a study on simulated data, we focus on how three different methods for inference influence the error in parameter estimation. We demonstrate both theoretically and empirically that our new method based on a solution of a general initial–boundary value problem for the Fick diffusion partial differential equation exhibits less bias and narrower confidence intervals of the estimated diffusion parameter, than two closed formula methods.

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Guanghao Jin, Young-Ju Lee, Hengguang Li Consider the Poisson equation with the Dirichlet boundary condition on a bounded convex polygonal domain Ω ⊂ R 2 . We investigate the finite element approximation of singular solutions that are due to the non-smoothness of the domain in the W p 1 norm ( 1 < p ≤ ∞ ). In particular, with analysis in weighted Sobolev spaces and weighted Hölder spaces, we provide regularity requirements on the given data and specific parameter-selection criteria for graded meshes, such that the resulting numerical approximation achieves the optimal convergence rate in W p 1 . Sample results from various numerical tests are provided to confirm the theory.

Abstract: Publication date: Available online 2 March 2017 Source:Computers & Mathematics with Applications Author(s): A. Raheem, Md. Maqbul In this paper, we established some sufficient conditions for oscillation of solutions of a class of impulsive partial fractional differential equations with forcing term subject to Robin and Dirichlet boundary conditions by using differential inequality method. As an application, we included an example to illustrate the main result.

Abstract: Publication date: 1 March 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 5 Author(s): Krzysztof Bartosz, David Danan, Paweł Szafraniec We study a fully dynamic thermoviscoelastic contact problem. The contact is assumed to be bilateral and frictional, where the friction law is described by a nonmonotone relation between the tangential stress and the tangential velocity. Weak formulation of the problem leads to a system of two evolutionary, possibly nonmonotone subdifferential inclusions of parabolic and hyperbolic type, respectively. We study both semidiscrete and fully discrete approximation schemes, and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the linear element solution. This theoretical result is illustrated numerically.

Abstract: Publication date: 1 March 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 5 Author(s): Pengyu Chen, Xuping Zhang, Yongxiang Li In this paper, we deal with a class of nonlinear time fractional non-autonomous evolution equations with delay by introducing the operators ψ ( t , s ) , φ ( t , η ) and U ( t ) , which are generated by the operator − A ( t ) and probability density function. The definition of mild solutions for studied problem was given based on these operators. Combining the techniques of fractional calculus, operator semigroups, measure of noncompactness and fixed point theorem with respect to k -set-contractive, we obtain new existence result of mild solutions with the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition and the closed linear operator − A ( t ) generates an analytic semigroup for every t > 0 . The results obtained in this paper improve and extend some related conclusions on this topic. At last, by utilizing the abstract result obtained in this paper, the existence of mild solutions for a class of nonlinear time fractional reaction–diffusion equation introduced in Ouyang (2011) is obtained.

Abstract: Publication date: 1 March 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 5 Author(s): Vasily E. Tarasov An exact discretization of fractional-order Laplacian for N -dimensional space is suggested. Particular solutions of fractional-order partial difference equations with the proposed discrete Laplace operators are suggested.

Abstract: Publication date: 1 March 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 5 Author(s): O.P. Porogo, B. Muatjetjeja, A.R. Adem In the present paper, we obtain a variational principle for a generalized coupled Zakharov–Kuznetsov system, which does not admit any Lagrangian formulation in its present form. The eminent Noether‘s theorem will then be employed to compensate for this approach. In addition, exact solutions will be constructed for the generalized coupled Zakharov–Kuznetsov system using the Kudryashov method and the Jacobi elliptic function method.

Abstract: Publication date: Available online 1 March 2017 Source:Computers & Mathematics with Applications Author(s): Yuan Zhou, Wen-Xiu Ma We apply the linear superposition principle to Hirota bilinear equations and generalized bilinear equations. By extending the linear superposition principle to complex field, we construct complex exponential wave function solutions first and then get complexions by taking pairs of conjugate parameters. A few examples of mixed resonant solitons and complexitons to Hirota and generalized bilinear differential equations are presented.

Abstract: Publication date: Available online 1 March 2017 Source:Computers & Mathematics with Applications Author(s): Benjamin Wassermann, Stefan Kollmannsberger, Tino Bog, Ernst Rank During the last ten years, increasing efforts were made to improve and simplify the process from Computer Aided Design (CAD) modeling to a numerical simulation. It has been shown that the transition from one model to another, i.e. the meshing, is a bottle-neck. Several approaches have been developed to overcome this time-consuming step, e.g. Isogeometric Analysis (IGA), which applies the shape functions used for the geometry description (typically B-Splines and NURBS) directly to the numerical analysis. In contrast to IGA, which deals with boundary represented models (B-Rep), our approach focuses on parametric volumetric models such as Constructive Solid Geometries (CSG). These models have several advantages, as their geometry description is inherently watertight and they provide a description of the models’ interior. To be able to use the explicit mathematical description of these models, we employ the Finite Cell Method (FCM). Herein, the only necessary input is a reliable statement whether an (integration-) point lies inside or outside of the geometric model. This paper mainly discusses such point-in-membership tests on various geometric objects like sweeps and lofts, as well as several geometric operations such as filleting or chamfering. We demonstrate that, based on the information of the construction method of these objects, the point-in-membership-test can be carried out efficiently and robustly.

Abstract: Publication date: Available online 1 March 2017 Source:Computers & Mathematics with Applications Author(s): Rehab M. El-Shiekh In this paper, a generalized variable-coefficient Boiti–Leon–Pempinlli (BLP) system is studied via the modified Clarkson and Kruskal (CK) direct reduction method connected with homogeneous balance (HB) method, which can describe the water waves in fluid physics. A direct similarity reduction to nonlinear ordinary differential system is obtained. By solving the reduced ordinary differential system, new analytical solutions (including solitary and periodic types) in terms of Jacobi elliptic functions are given for the variable-coefficient BLP system.

Abstract: Publication date: Available online 27 February 2017 Source:Computers & Mathematics with Applications Author(s): Rita Riedlbeck, Daniele A. Di Pietro, Alexandre Ern, Sylvie Granet, Kyrylo Kazymyrenko We derive equilibrated reconstructions of the Darcy velocity and of the total stress tensor for Biot’s poro-elasticity problem. Both reconstructions are obtained from mixed finite element solutions of local Neumann problems posed over patches of elements around mesh vertices. The Darcy velocity is reconstructed using Raviart–Thomas finite elements and the stress tensor using Arnold–Winther finite elements so that the reconstructed stress tensor is symmetric. Both reconstructions have continuous normal component across mesh interfaces. Using these reconstructions, we derive a posteriori error estimators for Biot’s poro-elasticity problem, and we devise an adaptive space–time algorithm driven by these estimators. The algorithm is illustrated on test cases with analytical solution, on the quarter five-spot problem, and on an industrial test case simulating the excavation of two galleries.

Abstract: Publication date: Available online 27 February 2017 Source:Computers & Mathematics with Applications Author(s): Rahman Farnoosh, Amirhossein Sobhani, Mohammad Hossein Beheshti In this paper, we introduce a new and considerably fast numerical method based on projection method in pricing discrete double barrier option. According to the Black–Scholes framework, the price of option in each monitoring dates is the solution of well-known partial differential equation that can be expressed recursively upon the heat equation solution. These recursive solutions are approximated by projection method and expressed in operational matrix form. The most important advantage of this method is that its computational time is nearly fixed against monitoring dates increase. Afterward, in implementing projection method we use Legendre polynomials as an orthogonal basis. Finally, the numerical results show the validity and efficiency of presented method in comparison with some others.

Abstract: Publication date: Available online 27 February 2017 Source:Computers & Mathematics with Applications Author(s): H. Liang, B.C. Shi, Z.H. Chai In this paper, an efficient phase-field-based lattice Boltzmann (LB) model with multiple-relaxation-time (MRT) collision operator is developed for the simulation of three-dimensional multiphase flows. This model is an extension of our previous two-dimensional model (Liang et al., 2014) to the three dimensions using the D3Q7 (seven discrete velocities in three dimensions) lattice for the Cahn–Hilliard equation and the D3Q15 lattice for the Navier–Stokes equations. Due to the less lattice-velocity directions used, the computational efficiency can be significantly improved in the study of three-dimensional multiphase flows, and simultaneously the present model can recover the Cahn–Hilliard equation and the Navier–Stokes equations correctly through the Chapman–Enskog procedure. We compare the present MRT model with its single-relaxation-time version and the previous LB model by using two benchmark interface-tracking problems, and numerical results show that the present MRT model can achieve a significant improvement in the accuracy and stability of the interface capturing. The developed model can also be able to deal with multiphase fluids with high viscosity ratio, which is demonstrated by the simulation of the layered Poiseuille flow and Rayleigh–Taylor instability at various viscosity ratios. The numerical results are found to be in good agreement with the analytical solutions or some available results. In addition, it is also found that the instability induces a more complex structure of the interface at a low viscosity.

Abstract: Publication date: Available online 27 February 2017 Source:Computers & Mathematics with Applications Author(s): Jan Březina, Pavel Exner The XFEM and Mortar methods can be used in combination with non-matching or non-conforming grids to deal with problems on complex geometries. However the information about the mesh intersection must be provided. We present algorithms for intersections between 1d and 2d unstructured multi component simplicial meshes and their intersections with a background unstructured 3d mesh. A common algorithm based on the advancing front technique is used for the efficient selection of candidate pairs among simplicial elements. Bounding interval hierarchy (BIH) of axes aligned bounding boxes (AABB) of elements is used to initialize the front tracking algorithm. The family of element intersection algorithms is built upon a line–triangle intersection algorithm based on the Plücker coordinates. These algorithms combined with the advancing front technique can reuse the results of calculations performed on the neighboring elements and reduce the number of arithmetic operations. Barycentric coordinates on each of the intersecting elements are provided for every intersection point. Benchmarks of the element intersection algorithms are presented and three variants of the global intersection algorithm are compared on the meshes raising from hydrogeological applications.

Abstract: Publication date: Available online 27 February 2017 Source:Computers & Mathematics with Applications Author(s): Yarong Zhang, Yinnian He, Hongbin Chen In the mathematical model of tumor growth, the initial boundary surface of the three dimensional tumor domain is known, but the tumor domain and its boundary surface change over time in a way that is unknown in advance. People are concerned whether the tumor is likely to spread or shrink and how the tumor will change. In this article, by the boundary element method, the free boundary problem in the three-dimensional tumor domain will be solved via the integration on a two-dimensional boundary surfaces (at the expense of singularity in the Green’s functions). We will numerically compute and graphically show the changing boundary surfaces of the three dimensional tumor domain over time. We will numerically analyze the trend of tumor growth with varying proliferation rate μ . Our numerical approach Professor Bei Hu proposed is new and our numerical experiments contribute to the prediction of tumor growth in clinical medicine.

Abstract: Publication date: Available online 24 February 2017 Source:Computers & Mathematics with Applications Author(s): Stéphane Abide, Belkacem Zeghmati This paper presents an analysis of a multigrid defect correction to solve a fourth-order compact scheme discretization of the Poisson’s equation. We focus on the formulation, which arises in the velocity/pressure decoupling methods encountered in computational fluid dynamics. Especially, the Poisson’s equation results of the divergence/gradient formulation and Neumann boundary conditions are prescribed. The convergence rate of a multigrid defect correction is investigated by means of an eigenvalues analysis of the iteration matrix. The stability and the mesh-independency are demonstrated. An improvement of the convergence rate is suggested by introducing the damped Jacobi and Incomplete Lower Upper smoothers. Based on an eigenvalues analysis, the optimal damping parameter is proposed for each smoother. Numerical experiments confirm the findings of this analysis for periodic domain and uniform meshes which are the working assumptions. Further numerical investigations allow us to extend the results of the eigenvalues analysis to Neumann boundary conditions and non-uniform meshes. The Hodge–Helmholtz decomposition of a vector field is carried out to illustrate the computational efficiency, especially by making comparisons with a second-order discretization of the Poisson’s equation solved with a state of art of algebraic multigrid method.

Abstract: Publication date: Available online 24 February 2017 Source:Computers & Mathematics with Applications Author(s): Hyung-Chun Lee, Max D. Gunzburger In this article, we consider an optimal control problem for an elliptic partial differential equation with random inputs. To determine an applicable deterministic control f ˆ ( x ) , we consider the four cases which we compare for efficiency and feasibility. We prove the existence of optimal states, adjoint states and optimality conditions for each cases. We also derive the optimality systems for the four cases. The optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space, respectively. The numerical experiments are performed for their efficiency and feasibility.

Abstract: Publication date: Available online 24 February 2017 Source:Computers & Mathematics with Applications Author(s): P. Constantinou, C. Xenophontos We consider a fourth order singularly perturbed boundary value problem posed in a square and the approximation of its solution by the h p version of the finite element method on the so-called Spectral Boundary Layer mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at an exponential rate when the error is measured in the energy norm. Our theoretical findings are illustrated through a numerical example.

Abstract: Publication date: Available online 24 February 2017 Source:Computers & Mathematics with Applications Author(s): Claire David The purpose of this work is to apply the results developed by Chemin and David (2013, 2015), to the Black–Scholes equation. This latter equation being directly linked to the heat equation, it enables us to propose a new approach allowing to control properties of the solution by means of a shape parameter.

Abstract: Publication date: Available online 24 February 2017 Source:Computers & Mathematics with Applications Author(s): David Colton Peter Monk has made numerous significant contributions to the field of inverse scattering theory. In the following I try to highlight Peter’s most significant achievements in this area with emphasis on the renaissance that took place in the mathematical and numerical treatment of inverse scattering problems that began in the mid 1980s.

Abstract: Publication date: Available online 23 February 2017 Source:Computers & Mathematics with Applications Author(s): M. Ghasemi The idea of differential quadrature is used to construct a new algorithm for the solution of differential equations. To determine the weighting coefficients of DQM, B-spline basis functions of degree r are used as test functions. The method is constructed on a set of points mixed from grid points and mid points of a uniform partition. Using the definition of B-splines interpolation as alternative, some error bounds are obtained for DQM. The method is successfully implemented on nonlinear boundary value problems of order m . Also the application of the proposed method to approximate the solution of multi-dimensional elliptic PDEs is included in the paper. As test problem, some examples of biharmonic and Poisson equations are solved in 2D and 3D. The results are compared with some existing methods to show the efficiency and performance of the proposed algorithm. Also some examples of time dependent PDEs are solved to compare the results with other existing spline based DQ methods.

Abstract: Publication date: Available online 21 February 2017 Source:Computers & Mathematics with Applications Author(s): Celia A.Z. Barcelos Variational methods for image registration involve minimizing a nonconvex functional with respect to the unknown displacement between two given images. In this paper, we present a new non-parametric image registration method posed as an optimization procedure, which combines a matching criterium and a smoothing term in an appropriate manner. Through the use of a weight function the model produces a smooth mapping between the two images, S and T , pointwise guided by the gradient variation of the target image S , the balance between the smoothing on the displacement vector field and the matching criterium allows for the occurrence of large deformations. We also present a framework for the fast linearized alternating direction method of multipliers (ADMM) for the numerical solution of the proposed model. The basic idea of the proposed algorithm is to incorporate an acceleration scheme into linearized ADMM. Experiments with both synthetic and real images in different domains illustrate that the proposed approach is efficient and effective.

Abstract: Publication date: Available online 21 February 2017 Source:Computers & Mathematics with Applications Author(s): Zhengguang Liu, Aijie Cheng, Hong Wang The computational work and memory requirement are bottlenecks for Galerkin finite element methods for peridynamic models because of their non-locality. In this paper, fast Galerkin and h p -Galerkin finite element methods are introduced and analyzed to solve a steady-state peridynamic model. We present a fast solution technique to accelerate non-square Toeplitz matrix–vector multiplications arising from piecewise-linear, piecewise-quadratic and piecewise-cubic Galerkin methods. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from O ( N 3 ) required by traditional methods to O ( N l o g 2 N ) and the memory requirement from O ( N 2 ) to O ( N ) without using any lossy compression, where N is the number of unknowns. The peridynamic model admits solutions having jump discontinuities. For problems with discontinuous solutions, we therefore introduce a piecewise-constant Galerkin method and give an h - and p -refinement algorithm. Then, we develop fast h p -Galerkin methods based on hybrid piecewise-constant/piecewise-linear and piecewise-constant/piecewise-quadratic finite element approximations. The new method reduces the computational work from O ( N 3 ) required by the traditional methods to O ( N l o g 2 N ) and the memory requirement from O ( N 2 ) to O ( N ) .

Abstract: Publication date: Available online 16 February 2017 Source:Computers & Mathematics with Applications Author(s): P.X. Yu, Z.F. Tian, Hongjie Zhang A rational high-order compact (RHOC) finite difference (FD) method on the nine-point stencil is proposed for solving the steady-state two-dimensional Navier–Stokes equations in the stream function–vorticity form. The resulting system of algebra equations can be solved by using the point-successive over- or under-relaxation (SOR) iteration. Numerical experiments, involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward-facing step flows, are carried out to validate the performance of the newly proposed method. Numerical solutions of the driven cavity problem with different grid mesh sizes (maximum being 513 × 513 ) for Reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature.

Abstract: Publication date: Available online 16 February 2017 Source:Computers & Mathematics with Applications Author(s): Farzaneh Hajabdollahi, Kannan N. Premnath Cascaded lattice Boltzmann method (LBM) involves the use of central moments in a multiple relaxation time formulation in prescribing the collision step. When the goal is to simulate low Mach number stationary flows, the greater the disparity between the magnitude of fluid speed and the sound speed, the higher is the numerical stiffness, which results in relatively large number of time steps for convergence of the LBM. One way to improve the steady state convergence of the scheme is to precondition the cascaded LBM, which reduces the disparities between the characteristic speeds or equivalently those between the eigenvalues of the system. In this paper we present a new preconditioned, two-dimensional cascaded LBM, where a preconditioning parameter is introduced into the equilibrium moments as well as the forcing terms. Particular focus is given to preconditioning differently the moments due to forcing at first and second orders so as to avoid any spurious effects in the emergent macroscopic equations. A Chapman–Enskog analysis performed on this approach shows consistency to the preconditioned Navier–Stokes equations. This modified central moment based scheme is then validated for accuracy by comparison against prior analytical or numerical results for certain benchmark problems, including those involving spatially variable body forces. Finally, significant steady state convergence acceleration of the preconditioned cascaded LBM is demonstrated for a set of characteristic parameters.

Abstract: Publication date: 15 February 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 4 Author(s): U Hou Lok, Yuh-Dauh Lyuu The implied volatility is the unique volatility value that makes the celebrated Black–Scholes formula yields a traded option’s price. Implied volatilities at varying strike prices and maturities form the implied volatility surface. Empirically, this surface is never flat. The local-volatility (LV) model is an option model that attempts to fit the implied volatility surface. It is popular because the preference freedom of the Black–Scholes model is retained. A tree consistent with the implied volatility surface generated by an LV model is known as implied tree. Past attempts to construct the implied tree, however, are prone to having invalid transition probabilities. In fact, this problem occurs even when the volatility surface is flat as in the Black–Scholes model. This paper unearths a potentially fundamental reason for that failure: the trees contain repelling fixed points. As efficient and valid trees for general LV models remain elusive despite decades of research, this paper turns to separable LV models. An efficient and valid binomial tree is then built for such models. Our novel tree is named the waterline tree because its upper part (the part that is above the water, so to speak) matches the moments of the price, whereas the lower part matches the moments of its logarithmic price. This break from traditional trees ensures that only attracting fixed points remain. Numerical results confirm the excellent performance of the waterline tree.

Abstract: Publication date: 15 February 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 4 Author(s): Khadijeh Baghaei This paper is concerned with the blow-up of solutions to a superlinear hyperbolic equation with linear damping term u t t − Δ u − ω Δ u t + μ u t = u p − 2 u , in [ 0 , T ] × Ω , where Ω ⊆ R n , n ≥ 1 , is a bounded domain with smooth boundary and T > 0 . Here, ω ≥ 0 and μ > − ω λ 1 , where λ 1 is the first eigenvalue of the operator − Δ under homogeneous Dirichlet boundary conditions. The recent results show that in the case of ω > 0 , if 2 < p < ∞ ( n = 1 , 2 ) or 2 < p ≤ 2 n n − 2 ( n ≥ 3 ) , then the solutions to the above equation blow up in a finite time under some suitable conditions on initial data. In this paper, in the case of ω > 0 , we obtain a lower bound for the blow-up time when the blow-up occurs. This result extends the recent results obtained by Sun et al. (2014) and Guo and Liu (2016).

Abstract: Publication date: 15 February 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 4 Author(s): Chenglin Li This paper is purported to investigate a ratio-dependent prey–predator system with cross-diffusion in a bounded domain under no flux boundary condition. The asymptotical stabilities of nonnegative constant solutions are investigated to this system. Moreover, without estimating the lower bounds of positive solutions, the existence, multiplicity of positive steady states are considered by using fixed points index theory, bifurcation theory, energy estimates and the differential method of implicit function and inverse function.

Abstract: Publication date: 15 February 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 4 Author(s): Rana D. Parshad, Emmanuel Quansah, Matthew A. Beauregard, Said Kouachi In Parshad et al. (2015) we showed that both the ODE and PDE versions of a classical three species food chain model, possess solutions that blow-up in finite time, for sufficiently large initial data. The object of the current work is to show that solutions to the PDE model can blow-up in finite time for small initial data as well. To the best of our knowledge, this is the first example of “diffusion induced” blow-up phenomenon for small initial data, in a three species system, where the corresponding ODE also has a possibility to blow-up for large initial data.

Abstract: Publication date: 15 February 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 4 Author(s): Huai Zhang, Tong Kang, Ran Wang, Yanfang Wang This paper is devoted to the study of a boundary data identification for an electromagnetic problem by means of the potential field method (the A - ϕ method). One part of the boundary is over-determined. The other part of the boundary is unreachable and has to be determined as a part of the problem. We design a constructive algorithm by the A - ϕ formulation to solve this problem. The numerical scheme is based on the steepest descent method (SDM) for the minimization of a regularized cost functional, having its derivative determined via an adjoint method. We analyze the properties of the cost functional and prove the convergence of the minimization process. The method is supported by several numerical experiments.

Abstract: Publication date: Available online 3 February 2017 Source:Computers & Mathematics with Applications Author(s): Chih-Ping Wu, Wei-Chen Li An asymptotic approach using the Eringen nonlocal elasticity theory and multiple time scale method is developed for the three-dimensional (3D) free vibration analysis of simply-supported, single-layered nanoplates and graphene sheets (GSs) embedded in an elastic medium. In the formulation, the small length scale effect is first introduced to the nonlocal constitutive equations by using a nonlocal parameter, then the mathematical processes of nondimensionalization, asymptotic expansion and successive integration are performed, and finally recurrent sets of motion equations for various order problems are obtained. The interactions between the nanoplates (or GSs) and their surrounding medium are modeled as a two-parameter Pasternak foundation. Nonlocal classical plate theory (CPT) is derived as a first-order approximation of the 3D nonlocal elasticity theory, and the motion equations for higher-order problems retain the same differential operators as those of nonlocal CPT, although with different nonhomogeneous terms. Some 3D nonlocal elasticity solutions of the natural frequency parameters of nanoplates (or GSs) with and without being embedded in the elastic medium and their corresponding through-thickness distributions of modal field variables are given to demonstrate the performance of the 3D asymptotic nonlocal elasticity theory.

Abstract: Publication date: Available online 25 January 2017 Source:Computers & Mathematics with Applications Author(s): Jing-Jing Hu, Chang-Feng Ma In this study, we consider the iteration solutions of the generalized Sylvester-conjugate matrix equation: A X B + C X ¯ D = E by a modified conjugate gradient method. When the system is consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial value given Hamiltonian matrix. Furthermore, we can get the minimum-norm solution X ∗ by choosing a special kind of initial matrix. Finally, some numerical examples are given to demonstrate the algorithm considered is quite effective in actual computation.

Abstract: Publication date: Available online 24 January 2017 Source:Computers & Mathematics with Applications Author(s): Yingjie Liang, Wen Chen, Belinda S. Akpa, Thomas Neuberger, Andrew G. Webb, Richard L. Magin Sephadex™ gel beads are commonly used to separate mixtures of similar molecules based on trapping and size exclusion from internal submicron diameter cavities. Water, as it freely moves through the porous gel and enclosed chambers of Sephadex™ beads, exhibits both normal (Gaussian) and anomalous (non-Gaussian) water diffusion. The apparent diffusion coefficient (ADC) of water in Sephadex™ gels can be measured using magnetic resonance imaging (MRI) by applying diffusion-weighted pulse sequences. This study investigates the relationship between the ADC of water and the complexity (i.e., size and number of cavities) of a series of Sephadex™ beads. We first classified the stochastic movement of water by using the solution to the space and time fractional diffusion equation to extract the ADC and the fractional time and space parameters ( α , β ), which are essentially the order of the respective fractional derivatives in Fick’s second law. From the perspective of the continuous time random walk (CTRW) model of anomalous diffusion, these parameters reflect waiting times (trapping) and jump increments (nano-flow) of the water in the gels. The observed MRI diffusion signal decay represents the Fourier transform of the diffusion propagator (i.e., the characteristic function of the stochastic process). In two series of Sephadex™ gel beads, we observed a strong inverse correlation between bead porosity (which is also responsible for molecular size exclusion) and the fractional order parameters; as the gels become more heterogeneous, the ADC decreases, both α and β are reduced and the diffusion exhibits anomalous (sub-diffusion) behavior. In addition, as a new measure for the structural complexity in Sephadex™ gel beads, we propose using the spectral and the cumulative spectral entropy that are derived from the observed characteristic function. We find that both measures of entropy increase with the porosity and tortuosity of the gel in a manner consistent with fractional order diffusional dynamics.

Abstract: Publication date: Available online 24 January 2017 Source:Computers & Mathematics with Applications Author(s): Ming Sun, Hongxing Rui In this paper, we present a coupling of a weak Galerkin method for the displacement of the solid phase with a standard mixed finite method for the pressure and velocity of the fluid phase in poroelasticity equation. Because our method provides a stable element combination for the mixed linear elasticity problem, it can avoid the poroelasticity locking mathematically. Indeed, uniform-in-time error estimates of all the unknowns are obtained for both semidiscrete scheme and fully discrete scheme without assuming that the constrained specific storage coefficient is uniformly positive. Compared with the method proposed by Jeonghun J. Lee, our method can get the arbitrary convergence order by changing degree of the polynomial. As for the computational cost, we can use the modified weak Galerkin method to eliminate the boundary term and reduce the number of unknowns. In addition, we do not use Gronwall inequality in the error analysis. Finally, the numerical experiments verify the theoretical analysis and show the effectiveness to overcome spurious pressure oscillations.

Abstract: Publication date: Available online 24 January 2017 Source:Computers & Mathematics with Applications Author(s): Ruizhi Yang, Ming Liu, Chunrui Zhang The dynamics of a diffusive toxin producing phytoplankton model with maturation delay and three-dimensional patch subject to Neumann boundary condition is investigated in this paper. The global stability of boundary equilibrium is obtained. The local stability of the coexistent equilibrium and the existence of Hopf bifurcation are investigated. The conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived.

Abstract: Publication date: Available online 24 January 2017 Source:Computers & Mathematics with Applications Author(s): Jesse Chan, Zheng Wang, Russell J. Hewett, T. Warburton We present a high order time-domain nodal discontinuous Galerkin method for wave problems on hybrid meshes consisting of both wedge and tetrahedral elements. We allow for vertically mapped wedges which can be deformed along the extruded coordinate, and present a simple method for producing quasi-uniform wedge meshes for layered domains. We show that standard mass lumping techniques result in a loss of energy stability on meshes of vertically mapped wedges, and propose an alternative which is both energy stable and efficient. High order convergence is demonstrated, and comparisons are made with existing low-storage methods on wedges. Finally, the computational performance of the method on Graphics Processing Units is evaluated.

Abstract: Publication date: Available online 23 January 2017 Source:Computers & Mathematics with Applications Author(s): Cung The Anh, Pham Thi Trang In this paper we consider the 3D Navier–Stokes–Voigt equations with periodic boundary conditions. We first prove the higher-order global regularity, including both Sobolev and Gevrey regularity, of solutions to the Navier–Stokes–Voigt equations. Then we show the convergence of solutions of the 3D Navier–Stokes–Voigt equations to the corresponding strong solution of the limit 3D Navier–Stokes equations on the interval of existence of the latter as the parameter tends to zero.

Abstract: Publication date: Available online 20 January 2017 Source:Computers & Mathematics with Applications Author(s): Huai-An Diao In this paper, we will study normwise, mixed and componentwise condition numbers for the linear mapping of the solution for general least squares with quadric inequality constraint (GLSQI) and its standard form (LSQI). We will introduce the mappings from the data space to the interested data space, and the Fréchet derivative of the introduced mapping can deduced through matrix differential techniques. Based on condition number theory, we derive the explicit expressions of normwise, mixed and componentwise condition numbers for the linear function of the solution for GLSQI and LSQI. Also, easier computable upper bounds for mixed and componentwise condition numbers are given. Numerical example shows that the mixed and componentwise condition numbers can tell us the true conditioning of the problem when its data is sparse or badly scaled. Compared with normwise condition numbers, the mixed and componentwise condition number can give sharp perturbation bounds.

Abstract: Publication date: Available online 20 January 2017 Source:Computers & Mathematics with Applications Author(s): Zhi Xia, Kui Du We consider the diffraction grating problem in optics, which has been modeled by a boundary value problem governed by a Helmholtz equation with transparent boundary conditions. A tensor-product finite element method is proposed to numerically solve the problem. An FFT-based matrix decomposition algorithm is developed to solve the linear system arising in the vertically layered medium case, which can be used as a preconditioning technique for the general case. Numerical examples are presented to illustrate the accuracy and efficiency of the method.