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 19 September 2018Source: Applied Numerical MathematicsAuthor(s): Liying Zhang, Lihai Ji In this paper, we consider stochastic Runge-Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for stochastic multi-symplecticity of stochastic Runge-Kutta methods. To present more clearly, we apply these ideas to three dimensional stochastic Maxwell equations driven by multiplicative noise, which play an important role in stochastic electromagnetism and statistical radiophysics areas. Theoretical analysis shows that the methods inherit the energy conservation law of the original system, and preserve the discrete stochastic multi-symplectic conservation law almost surely.

Abstract: Publication date: Available online 19 September 2018Source: Applied Numerical MathematicsAuthor(s): Jiyong Li Adapted Runge–Kutta–Nyström (ARKN) methods for solving oscillatory problems q″(t)+w2q(t)=f(q(t),q′(t)) have been investigated by several authors. Recently, Yang et al. [Applied Numerical Mathematics 58 (2008) 1375-1395] proposed trigonometrically-fitted ARKN (TFARKN) methods by introducing frequency depending coefficients into the terms in the internal stages. In applications the function f(q) often does not contain q′ explicitly and satisfies f(q)−w2q=−∇V(q) for some smooth function V(q). Then the problem can be considered as a separable Hamiltonian system. In this paper we investigate the symplecticity and symmetry of TFARKN methods for separable Hamiltonian systems and derive necessary and sufficient conditions for an TFARKN method to be symplectic and symmetric. Based on these conditions, two explicit symplectic and symmetric TFARKN method with order two and four, respectively, are constructed. Some numerical experiments are provided to confirm the theoretical expectations.

Abstract: Publication date: Available online 18 September 2018Source: Applied Numerical MathematicsAuthor(s): Yongtao Zhou, Chengjian Zhang In this paper, by combining the p-order block boundary value methods with the m-th Lagrange interpolation, a class of new numerical methods for solving nonlinear fractional differential equations with the γ-order (0

Abstract: Publication date: Available online 18 September 2018Source: Applied Numerical MathematicsAuthor(s): Yanping Chen, Haitao Leng, Wendi Yang In this paper, we study the error analysis of R-T mixed finite element methods for optimal control problems governed by stationary Stokes equations. To avoid the difficulty induced by the symmetry constraint on the stress tensor, we use the pseudostress proposed by Arnold and Falk [11] to replace it. Then, we obtain the pseudostress-velocity formulation of the optimal control problem. Moreover, we prove a priori and a posteriori error estimates for the new formulation, and an example is provided to confirm our results.

Abstract: Publication date: Available online 17 September 2018Source: Applied Numerical MathematicsAuthor(s): Linghua Kong, Meng Chen, Xiuling Yin In this paper, we construct a family of high order compact symplectic (S-HOC) schemes for the Klein-Gordon-Schrödinger (KGS) equation. The KGS can be cast into a Hamiltonian form. At first, we discretize the Hamiltonian system in space by a high order compact method which has higher convergent rate than general finite difference methods. Then the semi-discretized system is approximated in time by the Euler midpoint scheme which preserves the symplectic structure of the original system. The conserved quantities of the scheme, including symplectic structure conservation law, charge conservation law and energy conservation law, are discussed. The local truncation error and global error of the numerical solvers are investigated. Finally, some numerical verifications are presented to numerically validate the theoretical analysis. The numerical results are persuasive and illustrate the theoretical analysis.

Abstract: Publication date: Available online 13 September 2018Source: Applied Numerical MathematicsAuthor(s): M. Charina, M. Donatelli, L. Romani, V. Turati In this paper, motivated by applications to multigrid, we present families of anisotropic, bivariate, interpolating and approximating subdivision schemes. We study the minimality and polynomial generation/reproduction properties of both families and Hölder regularity of their prominent representatives. From the symbols of the proposed subdivision schemes, we define bivariate grid transfer operators for anisotropic multigrid methods. We link the generation/reproduction properties of subdivision to the convergence and optimality of the corresponding multigrid methods. We illustrate the performance of our subdivision based grid transfer operators on examples of anisotropic Laplacian and biharmonic problems.

Abstract: Publication date: Available online 13 September 2018Source: Applied Numerical MathematicsAuthor(s): Zi-Cai Li, Yimin Wei, Yunkun Chen, Hung-Tsai Huang In this paper, we study the Helmholtz equation by the method of fundamental solutions (MFS) using Bessel and Neumann functions. The bounds of errors are derived for bounded simply-connected domains, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Note that by using Bessel functions, the radius R of the source nodes is not necessarily to be larger than the maximal radius rmax of the solution domain. This is against the well-known rule: rmax

Abstract: Publication date: Available online 12 September 2018Source: Applied Numerical MathematicsAuthor(s): Min Zhong, Wei Wang In this paper, we propose a multilevel method for solving nonlinear inverse problems F(x)=y in Banach spaces. By minimizing the discretized version of the regularized functionals at different levels, we define a sequence of regularized approximations to the sought solution, which is shown to be stable and globally convergent. The penalty term Θ in regularized functionals is allowed to be non-smooth to include Lp−L1 or Lp−TV (Total Variation) reconstructions, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. Two parameter identification examples are presented to validate the theoretical analysis and to verify the effectiveness of the method.

Abstract: Publication date: Available online 11 September 2018Source: Applied Numerical MathematicsAuthor(s): Ali Lotfi In this paper an approximate scheme based on the combination of epsilon penalty method and an extended form of the well-known Ritz method is developed for solving a general class of fractional optimal control problems. The fractional derivative is in the Caputo sense, and the given fractional optimal control problem contains both state and control inequality constraints. The convergence of the method is discussed extensively and its effectiveness is verified by some illustrative test examples.

Abstract: Publication date: Available online 10 September 2018Source: Applied Numerical MathematicsAuthor(s): Ernesto Cáceres, Gabriel N. Gatica, Filánder A. Sequeira In this paper we introduce and analyze a mixed virtual element method (mixed-VEM) for a pseudostress-displacement formulation of the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. We follow a previous work by some of the authors, and employ a mixed formulation that does not require symmetric tensor spaces in the finite element discretization. More precisely, the main unknowns here are given by the pseudostress and the displacement, whereas other physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through simple postprocessing formulae in terms of the pseudostress variable. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, we utilize a well-known local projector onto a suitable polynomial subspace to define a calculable version of our discrete bilinear form, whose continuous version requires information of the variables on the interior of each element. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Babusˇka-Brezzi theory. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solutions as well as for the fully computable projections of them. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. In addition, this postprocessing formula can also be applied to the postprocessed stress tensor. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

Abstract: Publication date: Available online 5 September 2018Source: Applied Numerical MathematicsAuthor(s): Mirjam Walloth We present a new residual-type a posteriori error estimator for the discontinuous finite element solution of contact problems. The theoretical results are derived for two and three-dimensional domains and arbitrary gap functions. The estimator yields upper and lower bounds to a suitable error norm which measures the error in the displacements and in a quantity related to the contact stresses and the actual contact zone. In the derivation of the error estimator the local properties of the discontinuous solution are exploited appropriately so that, on the one hand, the error estimator has no contributions related to the non-linearity in the interior of the actual contact zone and, on the other hand, the critical region between the actual and non-actual contact zone can be well refined.

Abstract: Publication date: Available online 5 September 2018Source: Applied Numerical MathematicsAuthor(s): Liangliang Sun, Yun Zhang, Ting Wei In the present paper, we devote our effort to a nonlinear inverse problem for recovering a time-dependent potential term in a multi-term time-fractional diffusion equation from the boundary measured data. First we study the existence, uniqueness and regularity of solution for the direct problem by using the fixed point theorem. Then a stability estimate of inverse coefficient problem is obtained based on the regularity of solution of direct problem and some generalized Gronwall's inequalities. Numerically, we reformulate the inverse potential function into a variational problem, and we use a Levenberg-Marquardt method to find the approximate potential function. Numerical experiments for five examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.

Abstract: Publication date: Available online 3 September 2018Source: Applied Numerical MathematicsAuthor(s): Chieu Thanh Nguyen, B. Saheya, Yu-Lin Chang, Jein-Shan Chen In this paper, we explore a unified way to construct smoothing functions for solving the absolute value equation associated with second-order cone (SOCAVE). Numerical comparisons are presented, which illustrate what kinds of smoothing functions work well along with the smoothing Newton algorithm. In particular, the numerical experiments show that the well known loss function widely used in engineering community is the worst one among the constructed smoothing functions, which indicates that the other proposed smoothing functions can be employed for solving engineering problems.

Abstract: Publication date: Available online 3 September 2018Source: Applied Numerical MathematicsAuthor(s): Zhoufeng Wang, Rui Guo In this paper we consider the scattering of time-harmonic electromagnetic wave propagation in a homogeneous chiral environment by obstacles. The model is simplified to a two-dimensional scattering problem, and is formulated as a boundary value problem in a bounded domain by introducing the nonlocal boundary conditions associated with Dirichlet-to-Neumann operators. An a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place. The crucial part of the error analysis is to develop a duality argument and use the Bohren decomposition of the electromagnetic fields. The a posteriori error estimate consists of two parts, finite element approximation error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter. Numerical experiments are also presented to show the robustness and effectiveness of our numerical algorithm.

Abstract: Publication date: Available online 3 September 2018Source: Applied Numerical MathematicsAuthor(s): Manuel Baumann, Martin B. van Gijzen We consider wave propagation problems that are modeled in the frequency-domain, and that need to be solved simultaneously for multiple frequencies within a fixed range. For this, a single shift-and-invert preconditioner at a so-called seed frequency is applied. The choice of the seed is crucial for the performance of preconditioned multi-shift GMRES and is closely related to the parameter choice for the Complex Shifted Laplace preconditioner. Based on a classical GMRES convergence bound, we present an analytic formula for the optimal seed parameter that purely depends on the original frequency range. The new insight is exploited in a two-level preconditioning strategy: A shifted Neumann preconditioner with minimized spectral radius is additionally applied to multi-shift GMRES. Moreover, we present a reformulation of the multi-shift problem to a matrix equation solved with, for instance, global GMRES. Here, our analysis allows for rotation of the spectrum of the linear operator. Numerical experiments for the time-harmonic visco-elastic wave equation demonstrate the performance of the new preconditioners.

Abstract: Publication date: Available online 29 August 2018Source: Applied Numerical MathematicsAuthor(s): Xiaolong Zhang, John P. Boyd We revisit the spectral solution of the Thomas-Fermi problem for neutral atoms, urr−(1/r)u3/2=0 on r∈[0,∞] with u(0)=1 and u(∞)=0 to illustrate some themes in solving differential equations when there are complications, and also to make improvements in our earlier treatment. By “complications” we mean features of the problem that either destroy the exponential accuracy of a standard Chebyshev series, or render the classic Chebyshev approach inapplicable. The Thomas-Fermi problem has four complications: (i) a semi-infinite domain r∈[0,∞] (ii) a square root singularity in u(r) at the origin (iii) a fractional power nonlinearity and (iv) asymptotic decay as r→∞ that includes negative powers of r with fractional exponents. Our earlier treatment determined the slope at the origin to twenty-five decimal places, but no fewer than 600 basis functions were required to approximate a univariate solution that is everywhere monotonic, and all of the earlier tricks failed to recover an exponential rate of convergence in the truncation of the spectral series N, but only a high order convergence in negative powers of N. Here, using the coordinate z≡r to neutralize the square root singularity as before, we show that accuracy and rate of convergence are significantly improved by solving for the original unknown u(r) instead of the modified unknown v(r)=u(r) used previously. Without a further change of coordinate, a rational Chebyshev basis TLn(z;L) yields twelve decimal place for the slope at the origin, ur(0), with 70 basis functions and twenty-four places with a truncation N=100.True exponential accuracy can be restored by using an appropriate change of coordinates, z=G(Z), where G is some species of exponential. However, the various Chebyshev and Fourier series for the Thomas-Fermi function have “plural asymptotics”, that is, an∼aintermediate(n) for 1≪n≪...

Abstract: Publication date: Available online 28 August 2018Source: Applied Numerical MathematicsAuthor(s): Andrew Giuliani, Lilia Krivodonova We show that the theory for strong stability preserving (SSP) time stepping methods employed with the method of lines-type discretizations of hyperbolic conservation laws may result in overly stringent time step restrictions. We analyze a fully discrete finite volume method with slope reconstruction and a second order SSP Runge-Kutta time integrator to show that the maximum stable time step can be increased over the SSP limit. Numerical examples indicate that this result extends to two-dimensional problems on triangular meshes.

Abstract: Publication date: Available online 27 August 2018Source: Applied Numerical MathematicsAuthor(s): Claude Brezinski, K.A. Driver, Michela Redivo-Zaglia Corollary 2 in [1] states that for −32

Abstract: Publication date: Available online 25 August 2018Source: Applied Numerical MathematicsAuthor(s): Marco Berardi, Fabio Difonzo, Filippo Notarnicola, Michele Vurro Here some issues are studied, related to the numerical solution of Richards' equation in a one dimensional spatial domain by a technique based on the Transversal Method of Lines (TMoL). The core idea of TMoL approach is to semi-discretize the time derivative of Richards' equation: afterward a system of second order differential equations in the space variable is derived as an initial value problem.The computational framework of this method requires both Dirichlet and Neumann boundary conditions at the top of the column. The practical motivation for choosing such a condition is argued. We will show that, with the choice of the aforementioned initial conditions, our TMoL approach brings to solutions comparable with the ones obtained by the classical Methods of Lines (hereafter referred to as MoL) with corresponding standard boundary conditions: in particular, an appropriate norm is introduced for effectively comparing numerical tests obtained by MoL and TMoL approach and a sensitivity analysis between the two methods is performed by means of a mass balance point of view. A further algorithm is introduced for deducing in a self sustaining way the gradient boundary condition on top in the TMoL context.

Abstract: Publication date: Available online 25 August 2018Source: Applied Numerical MathematicsAuthor(s): M.A. Rincon, I.-S. Liu, W.R. Huarcaya, B.A. Carmo In this article, the error estimates for semi-discrete and totally discrete problems of a nonlinear model of elastic strings with moving boundary are established. We consider an extension of the Kirchhoff model, that takes into account the change of length of the string during vibration. The existence and uniqueness theorems of the problem, already known in the literature, will be stated for reference. The present numerical analysis is based on finite element method in spatial variable and finite difference method in time with the Newmark's approximation. Since the problem is nonlinear the resulting algebraic system is nonlinear to be solved by Newton's method.Numerical examples are presented for different kinds of moving boundary to verify the efficiency and feasibility of the method and check the coherence with the theoretical analysis. From the numerical results, the rate of convergence are shown to be consistent with the order of convergence expected from the theoretical ones.

Abstract: Publication date: Available online 23 August 2018Source: Applied Numerical MathematicsAuthor(s): Elias Jarlebring, Federico Poloni The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems x˙(t)=A0x(t)+A1x(t−τ)+B0u(t). We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unknown is the value U(τ/2)∈Rn×n, i.e., the delay Lyapunov matrix at time τ/2. This linear matrix equation with n2 unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the n2×n2 matrix associated to this linear system can be computed by solving a coupled matrix initial-value problem. A preconditioner for the iterative method is proposed based on solving a T-Sylvester equation MX+XTN=C, for which there are methods available in the literature. We prove that the preconditioner is effective under certain assumptions. The efficiency of the approach is illustrated by applying it to a time-delay system stemming from the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to n≈1000, i.e., a linear system with n2≈106 unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation.

Abstract: Publication date: Available online 21 August 2018Source: Applied Numerical MathematicsAuthor(s): Guang-Hui Zheng In this paper, we consider the backward problem introduced in [48] for Riesz–Feller fractional diffusion. To begin with, some basic properties of solution of the corresponding forward problem, such as the Lp estimates, symmetry property and asymptotic estimates, are established by Fourier analysis technique. And then, under various a priori bound assumptions, we give the L2 conditional stability estimates for the solution of backward problem and also its symmetry property. Moreover, in order to overcome the ill-posedness of the backward problem, we propose a new nonlocal regularization method (NLRM) to solve it. That is, the following nonlocal variational functional is introducedJ(φ)=12‖u(φ(x);x,T)−fδ(x)‖2+β2‖[Pα1⁎φ](x)‖2, where β∈(0,1) is a regularization parameter, “⁎” denotes the convolution operation and Pα1(x) is called a convolution kernel with parameter α1, which will be selected properly later. The minimizer of above variational problem is defined as the regularization solution, and the L2 estimates, symmetry property of regularization solution are given. These results actually show the well-posedness of nonlocal variational problem. Our idea is essentially that using this well-posed problem to approximate the backward (ill-posed) problem. Thus, under an a posteriori parameter choice rule, we deduce various convergence rate estimates under different a-priori bound assumptions for the exact solution. Finally, several numerical examples are given to show that the proposed numerical methods are effective and adaptive for different a-priori information.

Abstract: Publication date: Available online 20 August 2018Source: Applied Numerical MathematicsAuthor(s): Jehanzeb H. Chaudhry, John N. Shadid, Timothy Wildey This study considers adjoint based a posteriori estimation of the error in a quantity of interest computed from numerical solutions based on multistage implicit-explicit (IMEX) time integration schemes and an entropy-viscosity formulation for damped hyperbolic partial differential equations (PDEs). Hyperbolic systems are challenging to solve numerically due to the need to stabilize systems with discontinuous or nearly discontinuous solutions in an attempt to limit non-physical oscillations and provide accurate solutions. The goal of this effort is to provide error estimates based on adjoint operators, variational analysis and computable residuals. The error estimates quantify the total error as well as different contributions to the error arising from the time integration schemes and choices of numerical parameters in the numerical method.

Abstract: Publication date: Available online 17 August 2018Source: Applied Numerical MathematicsAuthor(s): Khalaf M. Alanazi, Zdzislaw Jackiewicz, Horst R. Thieme We extend our previous work on the spatial spread of fox rabies from one dimension to two dimensions. We consider the case when the latent period has fixed length. We use the method of lines to replace the spatial derivatives and the integral equations with algebraic approximations, then we apply the explicit continuous Runge-Kutta method of fourth order and discrete Runge-Kutta method of third order with six stages to numerically integrate the resulting systems of ordinary and delay differential equations. We discuss and confirm some of the major results we obtained in earlier work. The asymptotic speeds of spread observed in the two-dimensional simulations and in earlier work are discussed and compared with those found in nature.

Abstract: Publication date: Available online 15 August 2018Source: Applied Numerical MathematicsAuthor(s): Ying Xie, Chengjian Zhang This 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 Stynes A 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 Ding In 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 Shin A 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 Yang The 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 Zhang Given 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 Vong It 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 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 24 July 2018Source: Applied Numerical MathematicsAuthor(s): X. Claeys, V. Dolean, M.J. Gander Multi-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 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 18 July 2018Source: Applied Numerical MathematicsAuthor(s): Reza Mollapourasl, Majid Haghi, Ruihua Liu In 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 You In 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. Anh Fractional 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 Li In 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. Correa We 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: Available online 4 July 2018Source: Applied Numerical MathematicsAuthor(s): Argus A. Dunca Recent 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 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.