Abstract: Publication date: Available online 15 August 2018Source: Applied Numerical MathematicsAuthor(s): Ying Xie, Chengjian ZhangAbstractThis paper deals with nonlinear stochastic functional differential equations with piecewise continuous arguments (SFDEPCAs). Based on an adaptation to the underlying one-leg θ-methods for ODEs, a class of new one-parameter methods for nonlinear SFDEPCAs are introduced. The mean-square exponential stability criteria of analytical and numerical solutions are derived. Under the suitable conditions, it is proved that the one-parameter methods are convergent with strong order 1/2. Some numerical experiments are given to illustrate the theoretical results and computational advantages of the induced methods.

Abstract: Publication date: Available online 15 August 2018Source: Applied Numerical MathematicsAuthor(s): Chaobao Huang, Martin StynesAbstractA time-fractional reaction-diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω×[0,T], where Ω=(0,l)⊂R. The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t=0. Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretization on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L2(Ω) and H1(Ω) norms at each discrete time level tn by means of a non-trivial projection of the unknown solution into the finite element space. The L2(Ω) bound is optimal for all tn; the H1(Ω) bound is optimal for tn not close to t=0. An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp.

Abstract: Publication date: Available online 13 August 2018Source: Applied Numerical MathematicsAuthor(s): Hengfei DingAbstractIn this paper, we focus our attention on the development of the high-order numerical algorithm for the time-space tempered fractional diffusion-wave equation in two spatial dimensions. Based on the fourth-order fractional-compact difference operator, a new difference scheme with convergence order O(τ2+h14+h24) is derived, where τ is the temporal stepsize, h1 and h2 are the spatial stepsizes, respectively. The stability and convergence of the algorithm are investigated by the energy method and numerical experiment is carried out to verify the feasibility of the numerical algorithm.

Abstract: Publication date: Available online 11 August 2018Source: Applied Numerical MathematicsAuthor(s): Mi-Young Kim, Dong-wook ShinAbstractA discontinuous Galerkin method with Lagrange multipliers (DGLM) is developed to approximate the solution to the second-order elliptic problems. Lagrange multipliers for the solution and for the flux are considered on the edge/face of each element. The weak gradient and the weak divergence are defined for the elliptic problems. Lagrange multipliers for the solution and for the flux are shown to be the averages of the solutions and the “normal” fluxes at the edge/face, respectively. Unique solvability of the discrete system is proved and an error estimate is derived. The element unknowns are solved in terms of the Lagrange multipliers in element by element fashion. The Schur complement system of the Lagrange multipliers has a block structure, which is kept unchanged while the inside of the blocks gets dense in the higher order approximation. An explanation on algorithmic aspects is given. Some numerical results are presented.

Abstract: Publication date: Available online 9 August 2018Source: Applied Numerical MathematicsAuthor(s): Changqing YangAbstractThe Chebyshev polynomials and a collocation method are applied to the solution of the pantograph equation. A Chebyshev pantograph operational matrix is derived and used to reduce the pantograph equation to a system of algebraic equations. The convergence order of the proposed method is investigated in the L2-norm. Numerical examples are presented to verify the efficiency and accuracy of the proposed method. Results reveal that this method is accurate and easy to implement.

Abstract: Publication date: Available online 8 August 2018Source: Applied Numerical MathematicsAuthor(s): Zhe Li, Kai Zheng, Shugong ZhangAbstractGiven a set of distinct empirical points with uniform tolerance, based on the LDP algorithm proposed by Fassino and Torrente, we provide a verification algorithm that computes a polynomial, an admissible perturbed point set with verified error bound, such that the polynomial is guaranteed to vanish at a slightly admissible perturbed point set within computed error bound. The effectiveness of our algorithm is demonstrated in several examples.

Abstract: Publication date: Available online 24 July 2018Source: Applied Numerical MathematicsAuthor(s): Wen Li, Dongdong Liu, Seak-Weng VongAbstractIt is known that the spectral radius of the iterative tensor can be seen as an approximate convergence rate for solving multi-linear systems by tensor splitting iterative methods. So in this paper, first we give some spectral radius comparisons between two different iterative tensors. Then, we propose the preconditioned tensor splitting method for solving multi-linear systems, which provides an alternative algorithm with the choice of a preconditioner. In particular, also we give some spectral radius comparisons between the preconditioned iterative tensor and the original one. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned methods.

Abstract: Publication date: Available online 24 July 2018Source: Applied Numerical MathematicsAuthor(s): Iftikhar Ahmad, Hira IlyasAbstractIn 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 24 July 2018Source: Applied Numerical MathematicsAuthor(s): X. Claeys, V. Dolean, M.J. GanderAbstractMulti-trace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multi-trace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments.

Abstract: Publication date: Available online 23 July 2018Source: Applied Numerical MathematicsAuthor(s): Shenghao Li, Min ChenAbstractWe 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 18 July 2018Source: Applied Numerical MathematicsAuthor(s): Reza Mollapourasl, Majid Haghi, Ruihua LiuAbstractIn this paper, we consider European and American option pricing problems under regime switching jump diffusion models which are formulated as a system of partial integro-differential equations (PIDEs) with fixed and free boundaries. For free boundary problem arising in pricing American option, we use operator splitting method to deal with early exercise feature of American option. For developing a numerical technique we employ localized radial basis function generated finite difference (RBF-FD) approximation to overcome the ill-conditioning and high density issues of discretized matrices. The proposed method leads to linear systems with tridiagonal and diagonal dominant matrices. Also, in this paper the convergence and consistency of the proposed method are discussed. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options.

Abstract: Publication date: Available online 18 July 2018Source: Applied Numerical MathematicsAuthor(s): Qi Li, Liquan Mei, Bo YouAbstractIn this paper, we propose a second-order time accurate convex splitting scheme for the phase field crystal model. The temporal discretization is based on the second-order backward differentiation formula (BDF) and a convex splitting of the energy functional. The mass conservation, unconditionally unique solvability, unconditionally energy stability and convergence of the numerical scheme are proved rigorously. Mixed finite element method is employed to obtain the fully discrete scheme due to a sixth-order spatial derivative. Numerical experiments are presented to demonstrate the accuracy, mass conservation, energy stability and effectiveness of the proposed scheme.

Abstract: Publication date: Available online 17 July 2018Source: Applied Numerical MathematicsAuthor(s): S. Chen, F. Liu, I. Turner, V. AnhAbstractFractional differential equations have attracted considerable attention because of their many applications in physics, geology, biology, chemistry, and finance. In this paper, a two-dimensional Riesz space fractional diffusion equation on a convex bounded region (2D-RSFDE-CBR) is considered. These regions are more general than rectangle or circular domains. A novel alternating direction implicit method for the 2D-RSFDE-CBR with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the method are discussed. The resulting linear systems are Toeplitz-like and are solved by the preconditioned conjugate gradient method with a suitable circulant preconditioner. By the fast Fourier transform, the method only requires a computational cost of O(nlogn) per time step. These numerical techniques are used for simulating a two-dimensional Riesz space fractional FitzHugh-Nagumo model. The numerical results demonstrate the effectiveness of the method. These techniques can be extended to three spatial dimensions, which will be the topic of our future research.

Abstract: Publication date: Available online 11 July 2018Source: Applied Numerical MathematicsAuthor(s): Zhengguang Liu, Aijie Cheng, Xiaoli LiAbstractIn this article, we consider initial and boundary value problems for the diffusion-wave equation involving a Caputo fractional derivative(of order α, with 1

Abstract: Publication date: Available online 10 July 2018Source: Applied Numerical MathematicsAuthor(s): Adson M. Rocha, Juarez S. Azevedo, Saulo P. Oliveira, Maicon R. CorreaAbstractWe study the numerical approximation of functional integral equations, a class of nonlinear Fredholm-type integral equations of the second kind, by the collocation method with piecewise continuous basis functions. The resulting nonlinear algebraic system is solved with the Picard iteration method. Starting from the analysis of the continuous problem in L∞([a,b]), we prove the convergence of numerical solution and, under an additional regularity assumption, provide an a priori error estimate. Numerical examples illustrate the predicted theoretical results.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Mariam Al-Maskari, Samir KaraaAbstractWe consider the numerical approximation of a time-fractional cable equation involving two Riemann–Liouville fractional derivatives. We investigate a semidiscrete scheme based on the lumped mass Galerkin finite element method (FEM), using piecewise linear functions. We establish optimal error estimates for smooth and middly smooth initial data, i.e., v∈Hq(Ω)∩H01(Ω), q=1,2. For nonsmooth initial data, i.e., v∈L2(Ω), the optimal L2(Ω)-norm error estimate requires an additional assumption on mesh, which is known to be satisfied for symmetric meshes. A quasi-optimal L∞(Ω)-norm error estimate is also obtained. Further, we analyze two fully discrete schemes using convolution quadrature in time based on the backward Euler and the second-order backward difference methods, and derive error estimates for smooth and nonsmooth data. Finally, we present several numerical examples to confirm our theoretical results.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): R. Čiegis, O. SubočAbstractIn this paper we consider high-order compact finite difference schemes constructed on 1D non-uniform grids. We apply them to parabolic and Schrödinger equations. Stability of these schemes is investigated by using the spectral method. Computer experiments are applied in order to find critical grids for which the stability condition is violated. Such grids are obtained for the Schrödinger problem, but not for the parabolic problems. Numerical examples supporting our theoretical analysis are provided and discussed.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Merlin Fallahpour, Sean McKee, Ewa B. WeinmüllerAbstractOur aim is to simulate a nonlinear system of ODEs describing the flow in smectic liquid crystals. The nonlinear system is first linearized. We present a direct approach to compute the exact analytic solution of this linear system and use this solution as a starting profile in the Matlab package bvpsuite2.0 to obtain the approximate solution to the nonlinear system. Although, the solution of the nonlinear system has steep boundary layers and therefore is difficult to resolve, we demonstrate that bvpsuite2.0 can cope with the problem and provide an approximation with reasonable accuracy.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Jian Rong Loh, Abdulnasir Isah, Chang Phang, Yoke Teng TohAbstractIn this paper, we study the recently introduced Caputo and Fabrizio operator, which this new operator was derived by replacing the singular kernel in the classical Caputo derivative with the regular kernel. We introduce some useful properties based on the definition by Caputo and Fabrizio for a general order n

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Shi-Liang Wu, Peng GuoAbstractIn this paper, a class of modulus-based matrix splitting iteration methods for the quasi-complementarity problems is presented. The convergence analysis of the proposed methods is discussed. Numerical experiments show that the proposed methods are efficient.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Eugeniusz Zieniuk, Krzysztof SzerszeńAbstractThis paper proposes a modeling of the boundary at the interface between the body and the fluid for 2D flow problems described by the Stokes equation with NURBS curves. The theoretical representation of the boundary with the help of these curves is directly included into the classical boundary integral equations (BIE) for the Stokes equation. After this analytical inclusion of NURBS curves into a mathematical formula of the classical BIE, new generalized parametric integral systems (PIES) are obtained.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): G.A. Jarrad, A.J. RobertsAbstractFinite difference/element/volume methods of spatially discretising pdes impose a subgrid scale interpolation on the dynamics. In contrast, the so-called holistic discretisation approach developed herein constructs a natural subgrid scale field adapted to the whole system out-of-equilibrium dynamics. Consequently, the macroscale discretisation is systematically informed by the underlying microscale dynamics. We establish a new proof that there exists an exact closure of the spatially-discrete dynamics of a general class of reaction–advection–diffusion pdes. The approach also constructs new systematic approximations to the in-principle closure starting from a basis of simple, piecewise-linear, continuous approximation. Under inter-element coupling conditions that guarantee continuity of several field properties, the constructed holistic discretisation possesses desirable properties such as a natural cubic spline first-order approximation to the field, and the self-adjointness of the diffusion operator under periodic, Dirichlet and Neumann macroscale boundary conditions. As a concrete example, we demonstrate the holistic discretisation procedure on the well-known Burgers' pde, and compare the theoretical and numerical stability of the resulting discretisation to other approximations. The approach developed here promises to empower systematic construction of good, macroscale discretisations to a wide range of dissipative and wave pdes.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Tianliang Hou, Luoping Chen, Yin YangAbstractIn this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h=H2, the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): M.A. Zaky, E.H. Doha, J.A. Tenreiro MachadoAbstractA family of orthogonal systems of fractional functions is introduced. The proposed orthogonal systems are based on Jacobi polynomials through a fractional coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of fractional differential models. Approximation errors by the basic orthogonal projection are established. Three new kinds of fractional Jacobi–Gauss-type interpolations are introduced. As an example of application, an efficient approximation based on the proposed fractional functions to a fractional variational problem is presented and implemented. This approximation takes into account the potential irregularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. Implementation details are provided for the scheme, together with a series of numerical examples to show the efficiency of the proposed method.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Yiqun Li, Boying Wu, Melvin LeokAbstractIn this paper, we construct numerical schemes for spectral collocation methods and spectral variational integrators which converge geometrically. We present a systematic comparison of how spectral collocation methods and Galerkin spectral variational integrators perform in terms of their ability to reproduce accurate trajectories in configuration and phase space, their ability to conserve momentum and energy, as well as the linear stability of these methods when applied to some classical Hamiltonian systems.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): M.L.N. Gonçalves, F.R. OliveiraAbstractIn this paper, we propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate are established by using a general majorant condition. Two applications of such condition are provided: one is for functions whose derivatives satisfy a Hölder-like condition and the other is for functions that satisfy a Smale condition, which includes a substantial class of analytic functions. Some preliminary numerical experiments illustrating the applicability of the proposed method are also presented.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Nabil Chaabane, Vivette Girault, Beatrice Riviere, Travis ThompsonAbstractWe propose a stable element for the divergence operator that approximates the velocity by continuous linear polynomials plus piecewise constants and the pressure by piecewise constants. A uniform inf–sup condition is obtained for conforming meshes in two or three dimensions. The resulting method belongs to the class of enriched Galerkin methods, and is applied to the solution of a Stokes system. A priori error estimates in the energy norm and in the L2 norm are derived. Extensions to the Navier–Stokes system are presented.

Abstract: Publication date: October 2018Source: Applied Numerical Mathematics, Volume 132Author(s): Lukas EinkemmerAbstractThe automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used.In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion–advection equation, Burgers' equation with a reaction term, porous media equation, viscous Burgers' equation, Allen–Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): M. Braś, A. Cardone, Z. Jackiewicz, P. PierzchałaAbstractWe consider the class of implicit–explicit general linear methods (IMEX). Such schemes are designed for ordinary differential equation systems with right hand side function splitted into stiff and non-stiff parts. We investigate error propagation of IMEX methods up to the terms of order p+2. In addition, we construct IMEX schemes of order p and stage order q=p, p≤4 and we verify the performance of methods in several numerical experiments.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Mehdi Dehghan, Mostafa AbbaszadehAbstractThe main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6]. The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O(τ2−α). Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): S. Nemati, P. Lima, S. SedaghatAbstractIn this work, a spectral method based on a modification of hat functions (MHFs) is proposed to solve the fractional pantograph differential equations. Some basic properties of fractional calculus and the operational matrices of MHFs are utilized to reduce the considered problem to a system of linear algebraic equations. The greatest advantage of using MHFs is the large number of zeros in their operational matrix of fractional integration, product operational matrix and also pantograph operational matrix. This property makes these functions computationally attractive. Some illustrative examples are included to show the high performance and applicability of the proposed method and a comparison is made with the existing results. These examples confirm that the method leads to the results of convergence order O(h3).

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Ahmet Guzel, Catalin TrencheaAbstractThe explicit weakly-stable second-order accurate leapfrog scheme is widely used in the numerical models of weather and climate, in conjunction with the Robert–Asselin (RA) and Robert–Asselin–Williams (RAW) time filters. The RA and RAW filters successfully suppress the spurious computational mode associated with the leapfrog method, but also weakly damp the physical mode and degrade the numerical accuracy to first-order. The recent higher-order Robert–Asselin (hoRA) time filter reduces the undesired numerical damping of the RA and RAW filters and increases the accuracy to second up-to third-order. We prove that the combination of leapfrog-hoRA and Williams' step increases the stability by 25%, improves the accuracy of the amplitude of the physical mode up-to two significant digits, effectively suppresses the computational modes, and further diminishes the numerical damping of the hoRA filter.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Pouria Assari, Mehdi DehghanAbstractIn this investigation, a computational scheme is given to solve nonlinear one- and two-dimensional Volterra integral equations of the second kind. We utilize the radial basis functions (RBFs) constructed on scattered points by combining the discrete collocation method to estimate the solution of Volterra integral equations. All integrals appeared in the scheme are approximately computed by the composite Gauss–Legendre integration formula. The implication of previous methods for solving these types of integral equations encounters difficulties by increasing the dimensional of problems and sometimes requires a mesh generation over the solution region. While the new technique presented in the current paper does not increase the difficulties for higher dimensional integral equations due to the easy adaption of RBF and also needs no cell structures on the domains. Moreover, we obtain the error bound and the convergence rate of the proposed approach. Illustrative examples clearly show the reliability and efficiency of the method and confirm the theoretical error estimates.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Xiaoli Li, Hongxing RuiAbstractIn this article, a block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation with Neumann boundary condition is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δt1+σ/2+h2+k2+σ2) both for pressure and velocity are established on non-uniform rectangular grids, where Δt,h,k and σ are the step sizes in time, space in x- and y-direction, and distributed order. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Dongyang Shi, Huaijun YangAbstractA new low order nonconforming mixed finite element method (MFEM) is proposed and analyzed for time-fractional diffusion equation with element pair (CNRQ1+Q0×Q0). A new error estimate for the consistency error of nonconforming element CNRQ1 is proved, which leads to the superclose and superconvergence results of the original variable in broken H1 norm, and of the flux in L2 norm for a fully-discrete scheme with the Caputo derivative approximated by the classical L1 method. The results obtained herein improve the corresponding conclusions in the previous literature. Finally, some numerical results are provided to confirm the theoretical analysis.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Toshihiro YamadaAbstractThis paper shows a discretization method of solution to stochastic differential equations as an extension of the Milstein scheme. With a simple method, we reconstruct weak Milstein scheme through second order polynomials of Brownian motions without assuming the Lie bracket commutativity condition on vector fields imposed in the classical Milstein scheme and show a sharp error bound for it. Numerical example illustrates the validity of the scheme.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Z. Soori, A. AminataeiAbstractIn this paper, we propose a high-order scheme for the numerical solution of multi-term time fractional diffusion-wave (FDW) equation in one and two-dimensional on non-uniform grids. Based on the sixth-order non-uniform combined compact difference (NCCD) scheme in the space directions on non-uniform grids, an alternating direction implicit (ADI) method is proposed to split the equation into two separate one dimensional equations. The multi-term time fractional derivation is described in the Caputo's sense with scheme of order O(τ3−α), 1

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Denys Dutykh, Jean-Guy CaputoAbstractWe consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to be a 1D uniform line with end points identified with graph vertices. The interface conditions at these vertices are introduced and justified using conservation laws and an homothetic argument. We present a detailed methodology based on a symplectic finite difference scheme together with a special treatment at the junctions to solve the problem and apply it to the sine-Gordon equation. Numerical results on a simple graph containing four loops show the performance of the scheme for kinks and breathers initial conditions.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Xin Li, Luming ZhangAbstractIn this article, a trigonometric integrator sine pseudo-spectral (TISP) method is presented for the extended Fisher–Kolmogorov equation. This method depends on a Gautschi-type integrator in phase space to the temporal approximation and the sine pseudo-spectral method to the spatial discretization. Rigorous error estimates are carried out in the energy space by utilizing the mathematical induction. The error bound shows the new scheme which established by the TISP method has second-order accurate in time and spectral-order accurate in space. Moreover, the new scheme is generalized to higher dimensions. The compact finite difference (CFD) scheme in one and two dimensions which supported by the method of order reduction are constructed as a benchmark for comparisons. Comparison results between two schemes are given to confirm the theoretical studies and demonstrate the efficiency and accuracy of TISP method in both one and multi-dimensional problems.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): E. Aulisa, G. Bornia, S. Calandrini, G. CapodaglioAbstractIn this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs under no regularity assumptions on the solution of the problem. The proposed analysis provides three main contributions to the existing theory. The first novel contribution of this study is a convergence bound that depends on the number of multigrid smoothing iterations. This result is obtained under no regularity assumptions on the solution of the problem. A similar result has been shown in the literature for the cases of full regularity and partial regularity assumptions. Second, our theory applies to local refinement applications with arbitrary level hanging nodes. More specifically, for the smoothing algorithm we provide subspace decompositions that are suitable for applications where the multigrid spaces are defined on finite element grids with arbitrary level hanging nodes. Third, global smoothing is employed on the entire multigrid space with hanging nodes. When hanging nodes are present, existing multigrid strategies advise to carry out the smoothing procedure only on a subspace of the multigrid space that does not contain hanging nodes. However, with such an approach, if the number of smoothing iterations is increased, convergence can improve only up to a saturation value. Global smoothing guarantees an arbitrary improvement in the convergence when the number of smoothing iterations is increased. Numerical results are also included to support our theoretical findings.

Abstract: Publication date: September 2018Source: Applied Numerical Mathematics, Volume 131Author(s): Jing An, Jie Shen, Zhimin ZhangAbstractIn this paper we present and analyze a polynomial spectral-Galerkin method for nonlinear elliptic eigenvalue problems of the form −div(A∇u)+Vu+f(u2)u=λu,‖u‖L2=1. We estimate errors of numerical eigenvalues and eigenfunctions. Spectral accuracy is proved under rectangular meshes and certain conditions of f. In addition, we establish optimal error estimation of eigenvalues in some hypothetical conditions. Then we propose a simple iteration scheme to solve the underlying an eigenvalue problem. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results.

Abstract: Publication date: Available online 4 July 2018Source: Applied Numerical MathematicsAuthor(s): Argus A. DuncaAbstractRecent turbulence models such as the Approximate Deconvolution Model (ADM) or the Leray-deconvolution model are derived from the Navier–Stokes equations using the van Cittert approximate deconvolution method. As a consequence, the numerical error in the approximation of the Navier–Stokes weak solution with discrete solutions of the above models is influenced also by the discrete deconvolution error u−DNu‾h caused by the approximate deconvolution method. Here u is the flow field, u‾ is its average and DN is the N-th order van Cittert deconvolution operator.It is therefore important to analyze the deconvolution error u−DNu‾h in terms of the mesh size h, the filter radius α and the order N of the deconvolution operators used in the computation.This problem is investigated herein in the case of bounded domains and zero-Dirichlet boundary conditions. It is proved that on a sequence of quasiuniform meshes the L2 norm of the discrete deconvolution error convergences to 0 in the order of hk+1+KNh provided that the filter radius is in the order of the mesh size and the flow field has enough regularity. Here K

Abstract: Publication date: Available online 27 June 2018Source: Applied Numerical MathematicsAuthor(s): Runzhang Xu, Wei Lian, Xiangkun Kong, Yanbing YangAbstractThe 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 XuAbstractThis 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. TodorovAbstractWe 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. MalomedAbstractA 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. MarangellAbstractWe 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 MurashigeAbstractThis 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.

Abstract: Publication date: Available online 5 March 2018Source: Applied Numerical MathematicsAuthor(s): Youngmi Choi, Hyung-Chun LeeAbstractFinite element approximation solutions of the optimal control problems for stochastic Stokes equations with the forcing term perturbed by white noise are considered. To obtain the most efficient deterministic optimal control, we set up the cost functional as we proposed in [20]. Error estimates are established for the fully coupled optimality system using Green's functions and Brezzi–Rappaz–Raviart theory. Numerical examples are also presented to examine our theoretical results.

Abstract: Publication date: Available online 12 February 2018Source: Applied Numerical MathematicsAuthor(s): Haijian Yang, Feng-Nan HwangAbstractThis work aims to develop an adaptive nonlinear elimination preconditioned inexact Newton method as the numerical solution of large sparse multi-component partial differential equation systems with highly local nonlinearity. A nonlinear elimination algorithm used as a nonlinear preconditioner has been shown to be a practical technique for enhancing the robustness and improving the efficiency of an inexact Newton method for some challenging problems, such as the transonic full potential problems. The basic idea of our method is to remove some components causing troubles in order to decrease the impact of local nonlinearity on the global system. The two key elements of the method are the valid identification of the to-be-eliminated components and the choice of subspace correction systems, respectively. In the method, we employ the point-wise residual component of nonlinear systems as an indicator for selecting these to-be-eliminated components adaptively and build a subspace nonlinear system consisting of the components corresponding to the bad region and an auxiliary linearized subsystem to reduce the interfacial jump pollution. The numerical results demonstrate that the new approach significantly improves performance for incompressible fluid flow and heat transfer problems with highly local nonlinearity when compared to the classical inexact Newton method.

Abstract: Publication date: Available online 3 February 2018Source: Applied Numerical MathematicsAuthor(s): K.D. Brauss, A.J. MeirAbstractWe describe a parallel implementation for the numerical approximation of solutions to the three-dimensional viscous, resistive magnetohydrodynamics (MHD) equations using a velocity–current formulation. In comparison to other formulations, the velocity–current formulation presented in this paper is an integro-differential system of equations that incorporates nonideal boundaries and nonlinearities due to induction. The solution to the equations is approximated using a Picard iteration, discretized with the finite element method, and solved iteratively with the Krylov subspace method GMRES. Effective preconditioning strategies are required to numerically solve the resulting equations with Krylov solvers [12]. For GMRES convergence, the system matrix resulting from the discretization of the velocity–current formulation is preconditioned using a simple, block-diagonal Schur-complement preconditioner based on [14]. The MHD solver is implemented using freely available, well-documented, open-source, libraries deal.II, p4est, Trilinos, and PETSc, capable of scaling to tens of thousands of processors on state-of-the-art HPC architectures.

Abstract: Publication date: Available online 31 January 2018Source: Applied Numerical MathematicsAuthor(s): Yuhong Zhang, Haibiao Zheng, Yanren Hou, Li ShanAbstractIn this paper, we provide a coupled algorithm and a two-grid decoupled algorithm for a mixed Stokes–Stokes model, which is coupled by a nonlinear interface transmission condition. The coupled algorithm is to discretize the mixed model directly by standard finite element method. For the two-grid decoupled algorithm, we first solve the mixed model on a coarse grid, and update the solution on a fine grid by two separated Stokes problems. Under a hypothesis about the regularity of analytical solutions, optimal error estimates for two algorithms are achieved. Several numerical tests are given to verify our theoretical results.