Abstract: Publication date: Available online 11 January 2019Source: Applied Numerical MathematicsAuthor(s): Deepti Kaur, R.K. Mohanty In this work, compact difference scheme based on half-step discretization is proposed to solve fourth order time dependent partial differential equations subject to Dirichlet and Neumann boundary conditions. The difference method reported here is second order accurate in time and fourth order accurate in space. The scheme employs only three grid points at each time-level and the given boundary conditions are exactly satisfied with no further approximations at the boundaries. The linear stability of the presented method has been examined and it is shown that the proposed two-level finite difference method is unconditionally stable for a model linear problem. The developed method is directly applicable to fourth order parabolic partial differential equations with singular coefficients which is the main highlight of our work. The method is successfully tested on singular problem. The proposed method is applied to find the numerical solution of the classical nonlinear Kuramoto-Sivashinsky equation and the extended Fisher-Kolmogorov equation. Comparison of the obtained results with those of some earlier known methods demonstrate the superiority of the present approach.

Abstract: Publication date: Available online 8 January 2019Source: Applied Numerical MathematicsAuthor(s): Shounian Deng, Weiyin Fei, Yong Liang, Xuerong Mao We present a stochastic age-dependent population model that accounts for Markovian switching and variable delay. By using the approximate value at the nearest grid-point on the left of the delayed argument to estimate the delay function, we propose a class of split-step θ-method for solving stochastic delay age-dependent population equations (SDAPEs) with Markovian switching. We show that the numerical method is convergent under the given conditions. Numerical examples are provided to illustrate our results.

Abstract: Publication date: Available online 4 January 2019Source: Applied Numerical MathematicsAuthor(s): Xiaoli Li, Hongxing Rui, Shuangshuang Chen In this paper, the two MAC schemes are introduced and analyzed to solve the time fractional Stokes equation on non-uniform grids. One is the standard MAC scheme and another is the efficient MAC scheme, where the fast evaluation of the Caputo fractional derivative is used. The stability results are derived. We obtain the second order superconvergence in discrete L2 norm for both velocity and pressure. We also obtain the second order superconvergence for some terms of the H1 norm of the velocity on non-uniform grids. Besides, the efficient algorithm for the evaluation of the Caputo fractional derivative is used to save the storage and computation cost greatly. Finally, some numerical experiments are presented to show the efficiency and accuracy of MAC schemes.

Abstract: Publication date: Available online 3 January 2019Source: Applied Numerical MathematicsAuthor(s): Mohammed Elmustafa Amin, Xiangtuan Xiong We consider two inverse source problems for the radially symmetric and axis-symmetric heat conduction equations. The source terms depend on the space variables. By a transformation, the original inverse source problems can be transferred to nonlocal boundary problems with standard radially symmetric and axis-symmetric heat equations. Then the fundamental solution methods combined with the Tikhonov regularization method are provided to solve the nonlocal boundary problems. Once the solutions of the nonlocal boundary problems are found then the source terms can be recovered by numerical differentiation. Many numerical examples are shown to illustrate the performance of the proposed methods.

Abstract: Publication date: Available online 2 January 2019Source: Applied Numerical MathematicsAuthor(s): Yunhui Yin, Peng Zhu In this paper, the streamline-diffusion finite element method is applied to a two-dimensional convection-diffuse problem posed on the unit square, using a graded mesh of O(N2) points based on standard Lagrange polynomials of degree k≥1. We prove the method is convergent almost uniformly in the perturbation parameter ϵ, and obtains convergence order O(N−klogk+1(1ϵ)) in a streamline-diffusion norm under certain assumptions. Numerical experiments support the theoretical results.

Abstract: Publication date: Available online 22 December 2018Source: Applied Numerical MathematicsAuthor(s): Feng Liao, Luming Zhang, Tingchun Wang In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger-Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O(τ2+h4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis.

Abstract: Publication date: Available online 21 December 2018Source: Applied Numerical MathematicsAuthor(s): Tianliang Hou, Haitao Leng In this paper, we present a two-grid mixed finite element scheme for distributed optimal control problems governed by stationary Stokes equations. In order to avoid the difficulty caused by the symmetry constraint of the stress tensor, we use pseudostress to replace it. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We first prove that the difference between the interpolation and the numerical solution has superconvergence property for the control u with order h2. Then, using the postprocessing technique, we derive a second-order superconvergent result for the control u. Next, we construct a two-grid mixed finite element scheme and derive a priori error estimates. Finally, a numerical experiment is presented to verify the theoretical results.

Abstract: Publication date: Available online 17 December 2018Source: Applied Numerical MathematicsAuthor(s): Yu Zhang, Hai Bi, Yidu Yang In this paper, for biharmonic eigenvalue problems with clamped boundary condition in Rn which include plate vibration problem and plate buckling problem, we primarily study the two-grid discretization based on the shifted-inverse iteration of Ciarlet-Raviart mixed method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh πh can be reduced to the solution of an eigenvalue problem on a coarser mesh πH and the solution of a linear algebraic system on the fine mesh πh. With a new argument which is not covered by existing work, we prove that the resulting solution still maintains an asymptotically optimal accuracy when H>h≥O(H2). The surprising numerical results show the efficiency of our scheme.

Abstract: Publication date: Available online 13 December 2018Source: Applied Numerical MathematicsAuthor(s): A. Cordero, I. Giménez-Palacios, J.R. Torregrosa Searching zeros of nonlinear functions often employs iterative procedures. In this paper, we construct several families of iterative methods with memory from one without memory, that is, we have increased the order of convergence without adding new functional evaluations. The main aim of this manuscript yields in the advantage that the use of real multidimensional dynamics gives us to decide among the different classes designed and, afterwards, to select its most stable members. Moreover, we have found some elements of the family whose behavior includes strange attractors of different kinds that must be avoided in practice. In this sense, Feigenbaum diagrams have resulted an extremely useful tool. Finally, some of the designed classes with memory have been directly extended for solving nonlinear systems, getting an improvement in the efficiency in relation to other schemes with the same computational cost. These numerical tests confirm the theoretical results and show the good performance of the methods.

Abstract: Publication date: Available online 6 December 2018Source: Applied Numerical MathematicsAuthor(s): L. Zmour, A. Bouidi In this paper, we consider a numerical method for the White–Metzner model of viscoelastic fluid flows by a combination of the weighted least-squares (WLS) method and the streamline upwind/Petrov–Galerkin (SUPG) method. The constitutive equation is decoupled from the momentum and continuity equations, and the approximate solution is computed iteratively by solving the Stokes-like problem and a linearized constitutive equation using WLS and SUPG, respectively. The elastic viscous split stress formulation (EVSS) introduced in [1] is used for the discretization of the constitutive equation. An a priori error estimate for the WLS/SUPG method is derived and numerical results supporting the estimate are presented. We encounter no limit of the Weissenberg number for all values of ε satisfying 0

Abstract: Publication date: Available online 6 December 2018Source: Applied Numerical MathematicsAuthor(s): Shenglan Xie, Peng Zhu, Xiaoshen Wang In this paper, a fully discrete weak Galerkin (WG) finite element method is presented for solving time-dependent convection-dominated diffusion equations with convection term in non-conservation form. WG method and Crank–Nicolson scheme are used for spatial and temporal discretization, respectively. Stability and error analysis are established on polygonal mesh. Numerical examples are presented to verify our theoretical findings and show the efficiency of the method.

Abstract: Publication date: Available online 5 December 2018Source: Applied Numerical MathematicsAuthor(s): Florian Monteghetti, Denis Matignon, Estelle Piot This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss-Legendre quadrature rule is analyzed both theoretically and numerically. Numerical comparisons with both optimization-based and quadrature-based methods highlight its applicability. In addition, it is shown, on the example of a fractional delay differential equation, that quadrature-based discretization methods are spectrally correct, i.e. that they yield an unpolluted and convergent approximation of the essential spectrum linked to the fractional derivative, by contrast with optimization-based methods that can yield polluted spectra whose convergence is difficult to assess.

Abstract: Publication date: Available online 5 December 2018Source: Applied Numerical MathematicsAuthor(s): A. Ludu, A. Raghavendra Stable rotating polygonal hollow patterns were predicted theoretically and measured for Leidenfrost drops confined by circular boundary. Liquid free surface modes occur with different number of edges. This two-dimensional incompressible and inviscid flow is controlled by surface tension, evaporation reaction and gravitation. The model is based on shallow water theory and predicts the existence of sharp rotational polygonal waves as solitary waves solutions. These patterns may represent a signature of universality in shallow water rotating models since similar rotating hollow polygons are noticed in tropical cyclones, fast rotated liquids, Saturn's hexagon, and some plasma systems.

Abstract: Publication date: Available online 4 December 2018Source: Applied Numerical MathematicsAuthor(s): Rezvan Ghaffari, Farideh Ghoreishi In this paper, a reduced spline (RS) method based on a proper orthogonal decomposition (POD) technique for numerical solution of time fractional sub-diffusion equations is investigated. Combining POD with polynomial and non-polynomial spline methods yield a new model with lower dimensions and sufficiently high accuracy, so that the amount of computations and the calculation time decrease in comparison with usual spline methods. Although, the classical L1 scheme on uniform meshes is one of the most successful methods for approximation of the Caputo fractional derivative in time, due to the fact that the solution of time fractional sub-diffusion equations typically has a singularity at the origin, a weak convergence will conclude for such scheme. To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution. Also, the error estimates between the POD approximate solution of the RS scheme and the exact solution of fractional sub-diffusion equation are established for two types of uniform and graded meshes in time. Numerical examples are given to illustrate the feasibility and efficiency of the proposed method.

Abstract: Publication date: Available online 4 December 2018Source: Applied Numerical MathematicsAuthor(s): Seakweng Vong, Pin Lyu We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by further study on the generating function of the discretization matrix, second order convergence of the scheme is proved for diffusion coefficients satisfying a certain condition but are not necessary to be proportional. The theoretical results are justified by numerical tests.

Abstract: Publication date: Available online 4 December 2018Source: Applied Numerical MathematicsAuthor(s): M.J. Senosiain, A. Tocino The stochastic exponential method is applied to solve the undamped linear stochastic oscillator driven by a multidimensional Wiener process. The proposed additive noise model is new since the disturbances are introduced in the equivalent first order two dimensional system and can affect both space and velocity. It is shown that three important properties related to long time behavior of its analytical solution are preserved. In addition, the mean-square error of the numerical solution is analyzed. Numerical experiments confirming the theoretical results are presented.

Abstract: Publication date: Available online 30 November 2018Source: Applied Numerical MathematicsAuthor(s): Liwei Xu, Shangyou Zhang, George C. Hsiao In the finite element and boundary element coupling method, the Stekhlov-Poincaré mapping is often adopted at the coupling boundary, i.e., the unknown Neumann data are represented by the unknown Dirichlet data on the same boundary. This coupling method (either by a convolution integral, or by a Fourier series) is referred as the conventional method. We propose an alternative method, where the unknown Dirichlet data at the boundary are represented by the unknown solution value on an interior curve or surface, i.e., a Dirichlet-to-Dirichlet (D-to-D) mapping is employed instead of a Dirichlet-to-Neumann (D-to-N) mapping. The convergence of the method is shown to be of optimal order. 2D and 3D numerical experiments are performed to demonstrate the accuracy of the proposed method, comparing to the two conventional D-to-N coupling methods.

Abstract: Publication date: Available online 30 November 2018Source: Applied Numerical MathematicsAuthor(s): Sergey Charnyi, Timo Heister, Maxim A. Olshanskii, Leo G. Rebholz We study discretizations of the incompressible Navier-Stokes equations, written in the newly developed energy-momentum-angular momentum conserving (EMAC) formulation. We consider linearizations of the problem, which at each time step will reduce the computational cost, but can alter the conservation properties. We show that a skew-symmetrized linearization delivers the correct balance of (only) energy and that the Newton linearization conserves momentum and angular momentum, but conserves energy only up to the nonlinear residual. Numerical tests show that linearizing with 2 Newton steps at each time step is very effective at preserving all conservation laws at once, and giving accurate answers on long time intervals. The tests also show that the skew-symmetrized linearization is significantly less accurate. The tests also show that the Newton linearization of EMAC finite element formulation compares favorably to other traditionally used finite element formulation of the incompressible Navier-Stokes equations in primitive variables.

Abstract: Publication date: Available online 29 November 2018Source: Applied Numerical MathematicsAuthor(s): El Houcine Bergou, Serge Gratton, Jan Mandel Ensemble methods, such as the ensemble Kalman filter (EnKF), the local ensemble transform Kalman filter (LETKF), and the ensemble Kalman smoother (EnKS) are widely used in sequential data assimilation, where state vectors are of huge dimension. Little is known, however, about the asymptotic behavior of ensemble methods. In this paper, we prove convergence in Lp of ensemble Kalman smoother to the Kalman smoother in the large-ensemble limit, as well as the convergence of EnKS-4DVAR, which is a Levenberg-Marquardt-like algorithm with EnKS as the linear solver, to the classical Levenberg-Marquardt algorithm in which the linearized problem is solved exactly.

Abstract: Publication date: Available online 28 November 2018Source: Applied Numerical MathematicsAuthor(s): Mahboub Baccouch, Helmi Temimi, Mohamed Ben-Romdhane In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

Abstract: Publication date: Available online 28 November 2018Source: Applied Numerical MathematicsAuthor(s): Tanmay Sarkar, Praveen Chandrashekar We design and analyze a discontinuous Galerkin scheme for the initial-boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. The semi-discrete scheme is constructed by using the symmetrized version of the equation as introduced by Godunov. The resulting schemes are shown to be stable. Numerical experiments are performed in order to demonstrate the accuracy and convergence of the DG scheme through the L2-error and divergence error analysis. In the presence of discontinuities we add an artificial viscosity term to stabilize the solution.

Abstract: Publication date: Available online 28 November 2018Source: Applied Numerical MathematicsAuthor(s): Tianliang Hou, Wenzhu Jiang, Yueting Yang, Haitao Leng In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The gradient for the method belongs to the square integrable space instead of the classical H(div;Ω) space. P02-P1 mixed finite elements and Crank-Nicolson method are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze its convergence. It is shown that when the two mesh sizes satisfy h=H2, the two-grid method achieves the same convergence property as the P02-P1 mixed finite element method. Finally, we give a numerical example to verify the theoretical results.

Abstract: Publication date: Available online 22 November 2018Source: Applied Numerical MathematicsAuthor(s): A. Calini, C.M. Schober, M. Strawn We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schrödinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov–Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather.

Abstract: Publication date: Available online 22 November 2018Source: Applied Numerical MathematicsAuthor(s): G.M. Coclite, A. Fanizzi, L. Lopez, F. Maddalena, S.F. Pellegrino In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering because seems to provide an effective approach to modeling mechanical systems avoiding spatial discontinuous derivatives and body singularities. In particular, we will consider the linear model of peridynamics in a one-dimensional spatial domain. Here we will review some numerical techniques to solve this equation and propose some new computational methods of higher order in space; moreover we will see how to apply the methods studied for the linear model to the nonlinear one. Also a spectral method for the spatial discretization of the linear problem will be discussed. Several numerical tests will be given in order to validate our results.

Abstract: Publication date: Available online 22 November 2018Source: Applied Numerical MathematicsAuthor(s): Miroslav S. Petrov, Todor D. Todorov The paper deals with refinement techniques suitable for application of finite element multigrid methods with cubic trial functions. The 27-refinement strategy by Edelsbrunner and Grayson has not been studied from computational point of view up to now. This refinement strategy is said to be dark red refinement strategy (DRRS) by analogy with the red refinement strategy in the quadratic case. A detail analysis of DRRS is an aim of this paper. The dependence of the DRRS on the numbering of vertex nodes requires an algorithm development. Freudenthal partition of the cube has been widely used by researchers for obtaining hierarchical tetrahedral triangulations. In the cubic case, the refinement of the first Sommerville tetrahedron is of considerable practical importance. Since the DRRS generates one class of similarity only with a particular numbering of nodes, we have developed a detailed algorithm for proper implementation of this numbering. The situation is much more complicated when an arbitrary tetrahedron is refined. A large volume of computations are necessary in order to be established a correct numbering of the vertex nodes. If the best numbering of nodes is not chosen for all elements in all levels, the measure of degeneracy tends to infinity when the number of levels grows up unlimited. To avoid these difficulties a new canonical refinement strategy (CRS) is obtained. The CRS is superior compared to the 27-partition technique with respect to the degeneracy measure. It generates only regular tetrahedra and canonical simplices in all levels and for all mixed domains. These two tetrahedra are the most convenient from a computational point of view. Moreover, for all mixed domains the CRS essentially reduces the number of congruence classes. The new refinement strategy does not need an algorithm for correct numbering of the vertex nodes. The DRRS generates a very complicated refinement tree and a higher measure of degeneracy in combination with the face centered partition of the cube. On the contrary the Z refinement strategy obtained by the authors is superior than the DRRS in regard to the number of congruence classes and the measure of degeneracy. Some results on boundary value problems in changing domains are discussed. The advantages of the CRS are demonstrated by solving an anisotropic diffusion problem in a shrinking domain. The behavior of discretization and truncation errors in approximate finite element solutions is illustrated by varying sequences of finite element triangulations.

Abstract: Publication date: Available online 20 November 2018Source: Applied Numerical MathematicsAuthor(s): Zhongli Liu, Tianhai Tian, Hongjiong Tian In this paper, we propose an approach of combination of asymptotic and numerical techniques to solve highly oscillatory second-order initial value problems. An asymptotic expansion of the solution is derived in inverse of powers of the oscillatory parameter, which develops on two time scales, a slow time t and a fast time τ=ωt. The truncation with the first few terms of the expansion results in a very effective method of discretizing the highly oscillatory differential equation and becomes more accurate when the oscillatory parameter increases. Numerical examples show that our proposed asymptotic-numerical solver is efficient and accurate for highly oscillatory problems.

Abstract: Publication date: Available online 16 November 2018Source: Applied Numerical MathematicsAuthor(s): Zaid Odibat In this study, an optimal homotopy analysis approach will be described to deal with nonlinear fractional differential equations. The proposed approach presents a procedure to find an optimal auxiliary linear operator and the corresponding optimal initial approximation that will accelerate the convergence of series solutions for nonlinear differential equations with fractional derivatives. Then, a reliable modified version of the homotopy analysis method is presented to facilitate the calculations. Numerical comparison will be made to examine the computational efficiency and the pertinent features of the proposed algorithm. This algorithm is expected to be further employed to solve wide classes of nonlinear problems in fractional calculus.

Abstract: Publication date: Available online 14 November 2018Source: Applied Numerical MathematicsAuthor(s): Li Zhang, Minfu Feng, Jian Zhang In this work, we present and analyze a new weak Galerkin (WG) finite element method for the Brinkman equations. The finite element spaces which are made up of piecewise polynomials are easy to be constructed. Especially, the variational form considered in this work is based on two gradient operators. The stability, priori error estimates and L2 error estimates for velocity are proved in this paper. In addition, we prove that the new method also yields globally divergence-free velocity approximations. The convergence rates are independent of the Reynolds number, thus the new WG finite element method is efficient for both the Stokes and Darcy equations. Finally, numerical results illustrate the performance of the method and support the theoretical properties of the estimator.

Abstract: Publication date: Available online 9 November 2018Source: Applied Numerical MathematicsAuthor(s): Beibei Yuan, Mingwang Zhang, Zhengwei Huang We propose a wide neighborhood interior-point algorithm with arc-search for P⁎(κ) linear complementarity problem (LCP). Along the ellipsoidal approximation of the central path, the algorithm searches optimizers in every iteration. Assuming a strictly starting point is available, we show that the algorithm has O((1+2κ)(1+18κ)2nlog(x0)Ts0ε) iteration complexity. It matches the currently best known iteration bound for P⁎(κ) LCP. Some preliminary numerical results show that the proposed algorithm is efficient and reliable.

Abstract: Publication date: Available online 8 November 2018Source: Applied Numerical MathematicsAuthor(s): Xiaofeng Yang, Guo-Dong Zhang, Xiaoming He In this paper, we study a finite element approximation for a linear, first-order in time, unconditionally energy stable scheme proposed in [7] for solving the magneto-hydrodynamic equations. We first reformulate the semi-discrete scheme to the fully discrete version and then carry out a rigorous stability and error analysis for it. We show that the fully discrete scheme indeed leads to optimal error estimates for both velocity and magnetic field with some reasonable regularity assumptions. Moreover, under an alleviated time step constraint (δt≤1/ log(h) for 2D and δt≤h for 3D), the optimal error estimate for the pressure is derived as well.

Abstract: Publication date: Available online 7 November 2018Source: Applied Numerical MathematicsAuthor(s): Jingjun Zhao, Yu Li, Yang Xu In this paper, we construct a kind of product integration scheme for the nonlinear fractional ordinary differential equation. We design the scheme by considering the equivalent Volterra integral equation and using the idea of local Fourier expansion. We prove the high accuracy property of the new scheme and provide the stability analysis. Numerical experiments demonstrate the effectiveness of the scheme.

Abstract: Publication date: Available online 7 November 2018Source: Applied Numerical MathematicsAuthor(s): Ying Wang, Liquan Mei, Qi Li, Linlin Bu In this paper, we propose a highly accurate and conservative split-step spectral Galerkin (SSSG) scheme, which combines the standard second-order Strang split-step method for handling the nonlinear and the potential terms with the Legendre spectral Galerkin method for approximating the Riesz space-fractional derivatives, for the two-dimensional nonlinear space-fractional Schrödinger equation. The mass conservation property of the numerical solution is proved and the optimal error estimate with respect to spatial discretization is established by introducing an orthogonal projection operator. In addition, a matrix diagonalization technique is introduced to resolve the multi-dimensional difficulty and reduce the computational complexity in the implementation. Numerical experiments are presented to illustrate the accuracy and robustness of the SSSG method.

Abstract: Publication date: Available online 2 November 2018Source: Applied Numerical MathematicsAuthor(s): Hao Dong, Junzhi Cui, Yufeng Nie, Zihao Yang, Qiang Ma, Xiaohan Cheng In this paper, a novel multiscale computational method is presented for transient heat conduction problems of periodic porous materials with diverse periodic configurations in different subdomains. In these porous materials, heat transfer at microscale has an important impact on the macroscopic temperature field. Firstly, the second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on asymptotic homogenization method. Then, the error analysis in the pointwise sense is given to illustrate the importance of developing SOTS solutions. Furthermore, the error estimate for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method (FEM) and finite difference method (FDM). Finally, some numerical examples are shown, which demonstrate the feasibility and effectiveness of the SOTS numerical algorithm we proposed. In this paper, a unified two-scale computational framework is established for transient heat conduction problems of periodic porous materials with diverse periodic configurations in different subdomains.

Abstract: Publication date: Available online 2 November 2018Source: Applied Numerical MathematicsAuthor(s): Christopher A. Kennedy, Mark H. Carpenter Two new implicit–explicit, additive Runge–Kutta (ARK2) methods are given with fourth- and fifth-order formal accuracies, respectively. Both combine explicit Runge–Kutta (ERK) methods with explicit, singly-diagonally implicit Runge–Kutta (ESDIRK) methods and include an embedded method for error control. The two methods have ESDIRKs which are internally L-stable on stages three and higher, have only modestly negative eigenvalues to the stage and step algebraic-stability matrices and have stage-order two. To improve computational efficiency, the fourth-order method has a diagonal coefficient of 0.1235. This is done to offset much of the extra computational cost of an extra stage by facilitating iterative convergence at each stage. Linear stability domains for both ERK methods have been made quite large and the dominant coupling stability term between the stability of the ESDIRK and ERK for very stiff modes has been removed. Though the fourth-order method is one of the best all-around fourth-order IMEX ARK2 of which we are aware, the fifth-order method is likely best suited to mildly stiff problems with tight error tolerances. Methods are tested using the Van der Pol and Kaps' singular-perturbation problems. Results suggest that these new methods represent an improvement over existing methods of the same class.

Abstract: Publication date: Available online 30 October 2018Source: Applied Numerical MathematicsAuthor(s): Gabriel J. Lord, Antoine Tambue We consider the numerical approximation of the general second order semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. Our goal is to build two numerical algorithms with strong convergence rates higher than that of the standard semi-implicit scheme. In contrast to the standard time stepping methods which use basic increments of the noise, we introduce two schemes based on the exponential integrators, designed for finite element, finite volume or finite difference space discretisations. We prove the convergence in the root mean square L2 norm for a general advection diffusion reaction equation and a family of new Lipschitz nonlinearities. We observe from both the analysis and numerics that the proposed schemes have better convergence properties than the current standard semi-implicit scheme.

Abstract: Publication date: Available online 30 October 2018Source: Applied Numerical MathematicsAuthor(s): James Jackaman, Georgios Papamikos, Tristan Pryer We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation with periodic boundary conditions. We demonstrate that the scheme conserves energy up to solver tolerance. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.

Abstract: Publication date: Available online 29 October 2018Source: Applied Numerical MathematicsAuthor(s): Corrado Lattanzio, Corrado Mascia, Ramón G. Plaza, Chiara Simeoni This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen–Cahn equation –given byτ∂ttu+(1−τf′(u))∂tu−μ∂xxu=f(u), where τ,μ>0 and f is a cubic-like function with three zeros– represents a notable prototype.Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process, given by the substitution of the Fourier's law with the Maxwell–Cattaneo's law. A corresponding pseudo-kinetic scheme is also proposed for the Guyer–Krumhansl's law.For the Maxwell–Cattaneo case, numerical experiments1 are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves under a sign condition on the damping term g:=1−τf′. Moreover, important evidence of the validity of those results beyond the formal hypotheses g(u)>0 for any u is numerically established.

Abstract: Publication date: Available online 27 October 2018Source: Applied Numerical MathematicsAuthor(s): Deniz Bilman, Thomas Trogdon We compare the performance of well-known numerical time-stepping methods that are widely used to compute solutions of the doubly-infinite Fermi-Pasta-Ulam-Tsingou (FPUT) lattice equations. The methods are benchmarked according to (1) their accuracy in capturing the soliton peaks and (2) in capturing highly-oscillatory parts of the solutions of the Toda lattice resulting from a variety of initial data. The numerical inverse scattering transform method is used to compute a reference solution with high accuracy. We find that benchmarking a numerical method on pure-soliton initial data can lead one to overestimate the accuracy of the method.

Abstract: Publication date: Available online 26 October 2018Source: Applied Numerical MathematicsAuthor(s): Qingyi Zhan This article focuses on the existence of a true solution near a numerical approximate solution of random differential equations which can generate random dynamical systems. We prove a general finite-time shadowing theorem of random differential equations under some suitable assumptions and offer error bounds for shadowing distance based on some computable quantities. The application of this theorem is shown in the numerical simulations of chaotic orbits of the random Lorenz equations.

Abstract: Publication date: Available online 23 October 2018Source: Applied Numerical MathematicsAuthor(s): B.P. Moghaddam, J.A. Tenreiro Machado, M.L. Morgado The numerical solution of distributed order time fractional partial differential equations based on the midpoint quadrature rule and linear B-spline interpolation is studied. The proposed discretization algorithm follows the Du Fort-Frankel method. The unconditional stability and the convergence of the scheme are analyzed. Simulations illustrate the application of the new algorithm.

Abstract: Publication date: Available online 19 October 2018Source: Applied Numerical MathematicsAuthor(s): Zeting Liu, Fawang Liu, Fanhai Zeng In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is O(N−r+τ2), where N,τ,r are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids.

Abstract: Publication date: Available online 19 October 2018Source: Applied Numerical MathematicsAuthor(s): Mehdi Dehghan, Mostafa Abbaszadeh The MHD equation has some applications in physics and engineering. The main aim of the current paper is to propose a new numerical algorithm for solving the MHD equation. At first, the temporal direction has been discretized by the Crank-Nicolson scheme. Also, the unconditional stability and convergence of the time-discrete scheme have been investigated by using the energy method. Then, an improvement of element free Galerkin (EFG) i.e. the interpolating element free Galerkin method has been employed to discrete the spatial direction. Furthermore, an error estimate is presented for the full discrete scheme based on the Crank-Nicolson scheme by using the energy method. We prove that convergence order of the numerical scheme based on the new numerical scheme is O(τ2+δm). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. Numerical examples confirm the efficiency and accuracy of the proposed scheme.

Abstract: Publication date: Available online 15 October 2018Source: Applied Numerical MathematicsAuthor(s): Esmail Hesameddini, Mehdi Shahbazi In this paper, the Bernstein polynomials method is proposed for the numerical solution of a class of multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type. The main characteristic behind this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of Bernstein polynomials method for solving these equations are high, which will be shown by preparing some theorems. Also, the existence and uniqueness of solution have been proved. Finally, the numerical experiments are presented to show the excellent behavior and high accuracy of proposed scheme in comparison with some other well-known methods.

Abstract: Publication date: Available online 15 October 2018Source: Applied Numerical MathematicsAuthor(s): Sandra Carillo Third order nonlinear evolution equations, that is the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV) equation and other ones are considered: they all are connected via Bäcklund transformations. These links can be depicted in a wide Bäcklund Chart which further extends the previous one constructed in [22]. In particular, the Bäcklund transformation which links the mKdV equation to the KdV singularity manifold equation is reconsidered and the nonlinear equation for the KdV eigenfunction is shown to be linked to all the equations in the previously constructed Bäcklund Chart. That is, such a Bäcklund Chart is expanded to encompass the nonlinear equation for the KdV eigenfunctions [30], which finds its origin in the early days of the study of Inverse scattering Transform method, when the Lax pair for the KdV equation was constructed. The nonlinear equation for the KdV eigenfunctions is proved to enjoy a nontrivial invariance property. Furthermore, the hereditary recursion operator it admits [30] is recovered via a different method. Then, the results are extended to the whole hierarchy of nonlinear evolution equations it generates. Notably, the established links allow to show that also the nonlinear equation for the KdV eigenfunction is connected to the Dym equation since both such equations appear in the same Bäcklund chart.

Abstract: Publication date: Available online 10 October 2018Source: Applied Numerical MathematicsAuthor(s): A. Abdi, Z. Jackiewicz Various issues are discussed which are relevant to the development of the code for nonstiff differential systems based on a class of general linear methods with inherent Runge–Kutta stability.

Abstract: Publication date: Available online 4 October 2018Source: Applied Numerical MathematicsAuthor(s): Edouard Berthe, Duy-Minh Dang, Luis Ortiz-Gracia We present a robust and highly efficient Shannon wavelet pricing method for plain-vanilla foreign exchange European options under the jump-extended Heston model with multi-factor CIR interest rate dynamics. Under a Monte Carlo and partial differential equation hybrid computational framework, the option price can be expressed as an expectation, conditional on the variance factor, of a convolution product that involves the densities of the time-integrated domestic and foreign multi-factor CIR interest rate processes. We propose an efficient treatment to this convolution product that effectively results in a significant dimension reduction, from two multi-factor interest rate processes to only a single-factor process. By means of a state-of-the-art Shannon wavelet inverse Fourier technique, the resulting convolution product is approximated analytically and the conditional expectation can be computed very efficiently. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and efficiency of the method.

Abstract: Publication date: Available online 2 October 2018Source: Applied Numerical MathematicsAuthor(s): Evgueni Dinvay, Denys Dutykh, Henrik Kalisch In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties.In the current work, we review some of the existing fully dispersive systems, such as found in [1], [3], [8], [17], [23], [24]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

Abstract: Publication date: Available online 2 October 2018Source: Applied Numerical MathematicsAuthor(s): A. Alexandrou Himonas, Dionyssios Mantzavinos, Fangchi Yan The initial-boundary value problem on the half-line for the nonlinear Schrödinger equation supplemented with a Neumann boundary condition is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces Hs. The proof takes advantage of a novel explicit solution formula for the forced linear counterpart of the nonlinear problem, which is obtained via Fokas' unified transform method. This formula provides the basis for setting up a Picard iteration scheme and for deriving the linear estimates required for proving local well-posedness of the nonlinear problem via a contraction mapping argument.

Abstract: Publication date: Available online 1 October 2018Source: Applied Numerical MathematicsAuthor(s): Lothar Banz, Jan Petsche, Andreas Schröder In this paper various hybrid hp-finite element methods are discussed for elliptic model problems. In particular, a stabilized primal-hybrid hp-method is introduced which approximatively ensures continuity conditions across element interfaces and avoids the enrichment of the primal discretization space as usually required to fulfill some discrete inf-sup condition. A priori as well as a posteriori error estimates are derived for this method. The stabilized primal-hybrid hp-method is also applied to a model obstacle problem since it admits the definition of pointwise constraints. The paper also describes some extensions of primal, primal-mixed and dual-mixed methods as well as their hybridizations to hp-finite elements. In numerical experiments the convergence properties of the stabilized primal-hybrid hp-method are discussed and compared to all the other hp-methods introduced in this paper. The applicability of the a posteriori error estimates to drive h- and hp-adaptive schemes is also investigated.

Abstract: Publication date: Available online 27 September 2018Source: Applied Numerical MathematicsAuthor(s): Pratibha Shakya, Rajen Kumar Sinha This paper is concerned with residual type a posteriori error estimates of fully discrete finite element approximations for parabolic optimal control problems with measure data in a bounded convex domain. Two kinds of control problems, namely measure data in space and measure data in time, are considered and analyzed. We use piecewise linear functions for approximations of the state and co-state variables and piecewise constant functions for the control variable. The time discretization is based on the backward Euler implicit scheme. We derive a posteriori error estimates for the state, co-state and control variables in the L2(0,T;L2(Ω))-norm. Finally, numerical tests are presented to illustrate the performance of the estimators.

Abstract: Publication date: Available online 27 September 2018Source: Applied Numerical MathematicsAuthor(s): K. Maleknejad, M. Shahabi In this paper, an effective numerical technique based on the two-dimensional hybrid of Block-pulse functions and Legendre polynomials is presented to approximate the solution of a class of two-dimensional nonlinear Volterra integral equations of the second kind. The main idea of this work is to develop the operational matrices of hybrid functions into two-dimensional mode. These operational matrices are then applied to convert two-dimensional nonlinear integral equations into a system of nonlinear algebraic equations that can be solved numerically by Newton's method. An approximation of the error bound for the proposed method will be presented by proving some theorems. Finally, some numerical examples are considered to confirm the applicability and efficiency of the method, especially in cases where the solution is not sufficiently smooth.

Abstract: Publication date: Available online 24 July 2018Source: Applied Numerical MathematicsAuthor(s): Iftikhar Ahmad, Hira Ilyas In this paper, Homotopy Perturbation Method is applied to solve the nonlinear MHD Jeffery–Hamel arterial blood flow problem. Primarily, two-dimensional nonlinear Navier–Stokes equations have been converted into third order one-dimensional equation by means of transformation rule. Later the solution of governed equation is obtained by using Homotopy Perturbation Method. The proposed numerical results show a good agreement with reference solution for finite interval and emphasize to understand the human arterial blood flow rate. Further, accuracy and reliability of the proposed method is checked by increasing the iteration process up to third order. Finally, the results showed that product of angle between plates “α” and Reynolds number “Re” is directly proportional to the MHD Jeffery–Hamel flow.

Abstract: Publication date: Available online 23 July 2018Source: Applied Numerical MathematicsAuthor(s): Shenghao Li, Min Chen We prove the existence of a large family of two-dimensional diamond shaped standing waves for a Boussinesq system which describes two-way propagation of water waves in a channel. Our proof uses the Lyapunov-Schmidt method to find the bifurcation standing waves.

Abstract: Publication date: Available online 27 June 2018Source: Applied Numerical MathematicsAuthor(s): Runzhang Xu, Wei Lian, Xiangkun Kong, Yanbing Yang The main goal of this work is to investigate the initial boundary value problem of fourth order wave equation with nonlinear strain and logarithmic nonlinearity at three different initial energy levels, i.e., subcritical energy E(0)d. First, we prove the local existence of weak solution by Galerkin method. In the framework of potential well, we obtain the global existence and infinite time blow up of the solution with sub-critical initial energy. Moreover by the scaling technique, we obtain global existence and infinite time blow up of the solution with critical initial energy. Also, a high energy infinite time blow up result is established.

Abstract: Publication date: Available online 6 June 2018Source: Applied Numerical MathematicsAuthor(s): Yongbing Luo, Yanbing Yang, Md Salik Ahmed, Tao Yu, Mingyou Zhang, Ligang Wang, Huichao Xu This paper investigates the local existence, global existence and finite time blow up of the solution to the Cauchy problem for a class of nonlinear Klein–Gordon equation with general power-type nonlinearities. We give some sufficient conditions on the initial data such that the solution exists globally or blows up in finite time with low initial energy and critical energy. Further a finite time blow up result of the solution with high initial energy is proved.

Abstract: Publication date: Available online 30 May 2018Source: Applied Numerical MathematicsAuthor(s): V.S. Gerdjikov, M.D. Todorov We analyze the dynamical behavior of the N-soliton train in the adiabatic approximation of the perturbed nonlinear Schrödinger equation (NLSE) and the Manakov model. The perturbations include the simultaneous by a periodic external potential, and linear and nonlinear gain/loss terms. We derive the corresponding perturbed complex Toda chain (PCTC) models for both NLSE and Manakov model. We show that the soliton interactions dynamics for the PCTC models compares favorably to full numerical results of the original perturbed NLSE and Manakov model.

Abstract: Publication date: Available online 18 May 2018Source: Applied Numerical MathematicsAuthor(s): Bin Liu, Lu Li, Boris A. Malomed A class of periodic solutions of the nonlinear Schrödinger equation with non-Hermitian potentials are considered. The system may be implemented in planar nonlinear optical waveguides carrying an appropriate distribution of local gain and loss, in a combination with a photonic-crystal structure. The complex potential is built as a solution of the inverse problem, which predicts the potential supporting required periodic solutions. The main subject of the analysis is the spectral structure of the linear (in)stability for the stationary spatially periodic states in the periodic potentials. The stability and instability bands are calculated by means of the plane-wave-expansion method, and verified in direct simulations of the perturbed evolution. The results show that the periodic solutions may be stable against perturbations in specific Floquet–Bloch bands, even if they are unstable against small random perturbations.

Abstract: Publication date: Available online 17 May 2018Source: Applied Numerical MathematicsAuthor(s): P.N. Davis, P. van Heijster, R. Marangell We investigate the point spectrum associated with travelling wave solutions in a Keller–Segel model for bacterial chemotaxis with small diffusivity of the chemoattractant, a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We show that, for constant or sublinear consumption, there is an eigenvalue at the origin of order two. This is associated with the translation invariance of the model and the existence of a continuous family of solutions with varying wave speed. These point spectrum results, in conjunction with previous results in the literature, imply that in these cases the travelling wave solutions are absolute unstable if the chemotactic coefficient is above a certain critical value, while they are transiently unstable otherwise.

Abstract: Publication date: Available online 11 April 2018Source: Applied Numerical MathematicsAuthor(s): Sunao Murashige This paper describes a new type of long wave model for periodic internal waves propagating in permanent form at the interface between two immiscible inviscid fluids. This model for irrotational plane motion of these waves is derived in the complex velocity potential planes where the flow domains are conformally mapped. Since no smallness assumption of wave amplitude is made and the wave elevation at the interface is represented by a single-valued function of the velocity potential, this model is applicable to large-amplitude motions of which wave profile may overhang. Numerical examples demonstrate that the proposed model can produce overhanging solutions, and variations of solutions with wavelength or wave amplitude are qualitatively similar to those of the full Euler system. It is also pointed out that the kinematic condition at the interface is exactly satisfied in the proposed model for all wave amplitudes, but not in an existing long wave model derived in the physical plane.