A stabilized method for a secondary consolidation Biot's model

This work deals with the numerical solution of a secondary consolidation Biot's model. A family of finite difference methods on staggered grids in both time and spatial variables is considered. These numerical methods use a weighted two‐level discretization in time and the classical central difference discretization in space. A priori estimates and convergence results for displacements and pressure in discrete energy norms are obtained. Numerical examples illustrate the convergence properties of the proposed numerical schemes, showing also a non‐oscillatory behavior of the pressure approximation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Hybrid pseudospectral–finite difference method for solving a 3D heat conduction equation in a submicroscale thin film

This research aims to develop a time‐dependent pseudospectral‐finite difference scheme for solving a 3D dual‐phase‐lagging heat transport equation in a submicroscale thin film. The scheme uses periodic pseudospectral discretization in space and a fully second‐order finite difference discretization in time. The three consecutive time steps model is then solved explicitly, by using a preconditioned conjugate gradient method. The scheme is illustrated by an example which is used to investigate the heat transfer in a gold submicroscale thin film. Comparisons are made with available literature. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

High‐order energy‐preserving schemes for the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.     

Positive numerical solution for a nonarbitrage liquidity model using nonstandard finite difference schemes

In this article, we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonarbitrage liquidity model with observable parameters for derivatives. This nonlinear model considers that the parameters involved are observable from order book data. The proposed numerical method use a exact difference scheme in the linear convection‐reaction term, and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. To illustrate the accuracy of the method, the numerical results are compared with those produced by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 210‐221, 2014     

Analysis of a new BEM‐FEM coupling for two‐dimensional fluid‐solid interaction

In this work we present a new numerical method, based on a coupling of finite and boundary elements, to solve a fluid‐solid interaction problem in the plane. The discrete method uses classical Lagrange finite elements adapted to curved boundaries for the field variable and spectral approximation of the unknowns on the artificial boundary. We provide error estimates for this Galerkin scheme and propose a full discretization based on elementary quadrature formulae, showing that the perturbation due to numerical integration preserves the optimal rate of convergence. We also suggest an iterative method to solve the complicated linear systems arising from this type of schemes. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

A novel variable‐order fractional nonlinear Klein Gordon model: A numerical approach

In this article, a novel variable order fractional nonlinear Klein Gordon model is presented where the variable‐order fractional derivative is defined in the Caputo sense. The merit of nonstandard numerical techniques is extended here and we present the weighted average nonstandard finite difference method to study numerically the proposed model. Special attention is paid to study the convergence and to the stability analysis of the numerical technique. Moreover, the truncation error is analyzed. Three test examples are provided. Comparative studies are done between the used numerical technique and the weighted average finite difference method. It is found that the stability regions are larger by using the weighted average nonstandard finite difference method.     

Spectral graph wavelet optimized finite difference method for solution of Burger's equation with different boundary conditions

In this paper, spectral graph wavelet optimized finite difference method (SPGWOFD) has been proposed for solving Burger's equation with distinct boundary conditions. Central finite difference approach is utilized for the approximations of the differential operators and the grid on which the numerical solution is obtained is chosen with the help of spectral graph wavelet. Four test problems (with Dirichlet, Periodic, Robin and Neumann's boundary conditions) are considered and the convergence of the technique is checked. For assessing the efficiency of the developed technique, the computational time taken by the developed technique is compared to that of the finite difference method. It has been observed that developed technique is extremely efficient.     

Finite difference discretization of the Benjamin‐Bona‐Mahony‐Burgers equation

Numerical solutions of the Benjamin‐Bona‐Mahony‐Burgers equation in one space dimension are considered using Crank‐Nicolson‐type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L^∞‐norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin‐Bona‐Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

An almost L‐stable BDF‐type method for the numerical solution of stiff ODEs arising from the method of lines

A new BDF‐type scheme is proposed for the numerical integration of the system of ordinary differential equations that arises in the Method of Lines solution of time‐dependent partial differential equations. This system is usually stiff, so it is desirable for the numerical method to solve it to have good properties concerning stability. The method proposed in this article is almost L‐stable and of algebraic order three. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable some PDE problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A mixed collocation–finite difference method for 3D microscopic heat transport problems

Three-dimensional time-dependent initial-boundary value problems of a novel microscopic heat equation are solved by the mixed collocation–finite difference method in and on the boundaries of a particle when the thickness is much smaller than both the length and width. The collocation method on fixed grid size is used to approximate the space operator, whereas the finite difference scheme is used for time discretization. This new mixed method is applied to a novel heat problem in a particle, in order to compute the temperature distribution in and on the particle's surface. The second derivatives of the basis functions for the spectral approximation are derived. Direct substitution of derivatives in the model transforms the differential equation into a linear system of equations that is solved by the specific preconditioned conjugate gradient method. The high-order accuracy and resolution achieved by the proposed method allows one to obtain engineering-accuracy solution on coarse meshes. The consistency, stability and convergence analysis are provided and numerical results are presented.     

Finite difference spectral approximation for the time‐space fractional telegraph equation and its parameter estimation

The object of this paper is to present the numerical solution of the time‐space fractional telegraph equation. The proposed method is based on the finite difference scheme in temporal direction and Fourier spectral method in spatial direction. The fast Fourier transform (FFT) technique is applied to practical computation. The stability and convergence analysis are strictly proven, which shows that this method is stable and convergent with (2?α) order accuracy in time and spectral accuracy in space. Moreover, the Levenberg‐Marquardt (L‐M) iterative method is employed for the parameter estimation. Finally, some numerical examples are given to confirm the theoretical analysis.     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical results for the Klein‐Gordon equation in de Sitter spacetime

We consider the initial value problem for the Klein‐Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank‐Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible.     

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

In this paper, we study the numerical solution to time‐fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time‐fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Convergence of a difference scheme for the heat equation in a long strip by artificial boundary conditions

The numerical solution of the heat equation on a strip in two dimensions is considered. An artificial boundary is introduced to make the computational domain finite. On the artificial boundary, an exact boundary condition is proposed to reduce the original problem to an initial‐boundary value problem in a finite computational domain. A difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain. It is proved that the difference scheme is uniquely solvable, unconditionally stable and convergent with the convergence order 2 in space and order 3/2 in time in an energy norm. A numerical example demonstrates the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

In this work we construct and analyze some finite difference schemes used to solve a class of time‐dependent one‐dimensional convection‐diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank‐Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples which illustrate the theoretical results, in the case of using the Euler implicit method, and give better numerical behaviour than that predicted theoretically, showing order two in time and order N^?2log²N in space, if the Crank‐Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A‐stable SDIRK with two stages and a third‐order HODIE difference scheme, showing its uniformly convergent behavior, reaching order three, up to a logarithmic factor. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

The Global Behavior of Finite Difference-Spatial Spectral Collocation Methods for a Partial Integro-Differential Equation with a Weakly Singular Kernel

The $z$-transform is introduced to analyze a full discretization method for a partial integro-differential equation (PIDE) with a weakly singular kernel. In this method, spectral collocation is used for the spatial discretization, and, for the time stepping, the finite difference method combined with the convolution quadrature rule is considered. The global stability and convergence properties of complete discretization are derived and numerical experiments are reported.     

解非线性有限元方程的逐层校正迭代法

本文提出一种求解非线性有限元方程的逐层校正迭代法.有关数值分析表明,当网格分划较细,网格分划参数h_j较小时,仅需一次简单的迭代和校正步骤就可满足数值计算的要求,使用该方法的计算复杂性是最佳阶的,即为O(N_j),其中N_j为最细网格层上离散结点变量的数目.     

Mixed finite volume method for nonlinear elliptic problems

In this article we construct and analyze a mixed finite volume method for second‐order nonlinear elliptic problems employing H(div; Ω)‐conforming approximations for the vector variable and completely discontinuous approximations for the scalar variable. The main attractive feature of our method is that, although the vector variable is H(div; Ω)‐conforming, one can eliminate it in a local manner to obtain a discontinuous Galerkin method for the scalar variable. Optimal error estimates will be established for both vector and scalar variables. We also present a fully discrete version of this method that is more convenient for computational purposes. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005