Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data

In this article, we develop several first order fully discrete Galerkin finite element schemes for the Oldroyd model and establish the corresponding stability results for these numerical schemes with smooth and nonsmooth initial data. The stable mixed finite element method is used to the spatial discretization, and the temporal treatments of the spatial discrete Oldroyd model include the first order implicit, semi‐implicit, implicit/explicit, and explicit schemes. The ‐stability results of the different numerical schemes are provided, where the first‐order implicit and semi‐implicit schemes are the ‐unconditional stable, the implicit/explicit scheme is the ‐almost unconditional stable, and the first order explicit scheme is the ‐conditional stable. Finally, some numerical investigations of the ‐stability results of the considered numerical schemes for the Oldroyd model are provided to verify the established theoretical findings.     

Analysis of second order and unconditionally stable BDF2‐AB2 method for the Navier‐Stokes equations with nonlinear time relaxation

In this study, we first consider a second order time stepping finite element BDF2‐AB2 method for the Navier‐Stokes equations (NSE). We prove that the method is unconditionally stable and accurate. Second, we consider a nonlinear time relaxation model which consists of adding a term “” to the Navier‐Stokes Equations with the algorithm depends on BDF2‐AB2 method. We prove that this method is unconditionally stable, too. We applied the BDF2‐AB2 method to several numeral experiments including flow around the cylinder. We have also applied BDF2‐AB2 method with nonlinear time relaxation to some problems. It is observed that when the equilibrium errors are high, applying BDF2‐AB2 with nonlinear time relaxation method to the problem yields lower equilibrium errors.     

High‐order compact difference schemes for the modified anomalous subdiffusion equation

In this article, two kinds of high‐order compact finite difference schemes for second‐order derivative are developed. Then a second‐order numerical scheme for a Riemann–Liouvile derivative is established based on a fractional centered difference operator. We apply these methods to a fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability, and convergence analysis of these difference schemes are studied by using Fourier method. The convergence orders of these numerical schemes are and , respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 213–242, 2016     

A block‐centered finite difference method for fractional Cattaneo equation

In this article, a block‐centered finite difference method for fractional Cattaneo equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norm with optimal order of convergence both for pressure and velocity are established on nonuniform rectangular grids. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.     

Compact finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly singular kernel

In this paper, we develop a high‐order finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly singular kernel. The fractional derivative is used in the Riemann‐Liouville sense. We prove the unconditional stability and convergence of scheme using energy method and show that the convergence order is . We provide some numerical experiments to confirm the efficiency of suggested scheme. The results of numerical experiments are compared with analytical solutions to show the efficiency of proposed scheme. It is illustrated that the numerical results are in good agreement with theoretical ones.     

Fourth‐order compact difference schemes for 1D nonlinear Kuramoto–Tsuzuki equation

In this article, first, we establish some compact finite difference schemes of fourth‐order for 1D nonlinear Kuramoto–Tsuzuki equation with Neumann boundary conditions in two boundary points. Then, we provide numerical analysis for one nonlinear compact scheme by transforming the nonlinear compact scheme into matrix form. And using some novel techniques on the specific matrix emerged in this kind of boundary conditions, we obtain the priori estimates and prove the convergence in norm. Next, we analyze the convergence and stability for one of the linearized compact schemes. To obtain the maximum estimate of the numerical solutions of the linearized compact scheme, we use the mathematical induction method. The treatment is that the convergence in norm is obtained as well as the maximum estimate, further the convergence in norm. Finally, numerical experiments demonstrate the theoretical results and show that one of the linearized compact schemes is more accurate, efficient and robust than the others and the previous. It is worthwhile that the compact difference methods presented here can be extended to 2D case. As an example, we present one nonlinear compact scheme for 2D Ginzburg–Landau equation and numerical tests show that the method is accurate and effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2080–2109, 2015     

Higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations

In this article, we develop a higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. We approximate the retarded terms of the model problem using Taylor's series expansion and the resulting time‐dependent singularly perturbed problem is discretized by the implicit Euler scheme on uniform mesh in time direction and a special hybrid finite difference scheme on piecewise uniform Shishkin mesh in spatial direction. We first prove that the proposed numerical discretization is uniformly convergent of , where and denote the time step and number of mesh‐intervals in space, respectively. After that we design a Richardson extrapolation scheme to increase the order of convergence in time direction and then the new scheme is proved to be uniformly convergent of . Some numerical tests are performed to illustrate the high‐order accuracy and parameter uniform convergence obtained with the proposed numerical methods.     

Numerical analysis and iteration acceleration of a fully implicit scheme for nonlinear diffusion problem with second‐order time evolution

Fully implicit schemes with second‐order time evolutions have been applied to simulate nonlinear diffusion problems precisely for a long time, but there is seldom theoretical study for either their convergence properties or efficient iterations. Here, a second‐order time evolution fully implicit scheme for two‐dimensional nonlinear divergence diffusion problem is analyzed. The unique existence of its solution is given. Two new methods are provided to prove its convergence, including entire inductive hypothesis reasoning and a two‐step reasoning process. Rigorous analysis shows the scheme is stable; its solution has second‐order convergence in both space and time to the exact solution of the problem. The convergence is applied to analyze a Newton iteration accelerating the computation and show its quadratic convergent speed and second‐order accuracy. The reasoning techniques also adapt to first‐order time accuracy schemes, and can be extended to analyze a wide class of nonlinear schemes for nonlinear problems. Numerical tests highlight the theoretical results and demonstrate the high performance of the algorithms. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 121–140, 2016     

Conservative compact difference schemes for the coupled nonlinear schrödinger system

In this article, some conservative compact difference schemes are explored for the strongly coupled nonlinear schrödinger system. After transforming the scheme into matrix form, we prove the existence and uniqueness, convergence and stability of the difference solutions for one nonlinear scheme in the norm by using some techniques of matrix theory. Numerical results show that one nonlinear scheme is the most efficient of all the compact schemes constructed here. It allows much larger time steps than the others. The second most efficient compact scheme is a linear one. We then give numerical simulations to two soliton interactions for the two most efficient compact schemes. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 749–772, 2014     

Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

In this article, we obtain local energy and momentum conservation laws for the Klein‐Gordon‐Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy‐ and momentum‐preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time‐space region. With suitable boundary conditions, the schemes will be charge‐ and energy‐/momentum‐preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329–1351, 2017     

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

In this study, with the aid of Wolfram Mathematica 11, the modified exp ‐expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well‐known nonlinear evolutionary equation, namely; the two‐component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two‐component second order KdV evolutionary system with the finite forward difference method by using the Fourier‐Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of and norm error. We present the comparison between the exact and numerical solutions of the two‐component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp ‐expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models.     

An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience

A class of time‐dependent singularly perturbed convection‐diffusion problems with retarded terms arising in computational neuroscience is considered. In particular, a numerical scheme for the parabolic convection‐diffusion problem where the second‐order derivative with respect to the spatial direction is multiplied by a small perturbation parameter and the shifts are of is constructed. The Taylor series expansion is used to tackle the shift terms. The continuous problem is semidiscretized using the Crank‐Nicolson finite difference method in the temporal direction and the resulting set of ordinary differential equations is discretized using a midpoint upwind finite difference scheme on an appropriate piecewise uniform mesh, which is dense in the boundary layer region. It is shown that the proposed numerical scheme is second‐order accurate in time and almost first‐order accurate in space with respect to the perturbation parameter . To validate the computational results and efficiency of the method some numerical examples are encountered and the numerical results are compared with some existing results. It is observed that the numerical approximations are fairly good irrespective of the size of the delay and the advance till they are of . The effect of the shifts on the boundary layer has also been observed.     

Unconditional superconvergence analysis of an H1‐galerkin mixed finite element method for nonlinear Sobolev equations

An efficient H¹‐Galerkin mixed finite element method (MFEM) is presented with and zero order Raviart‐Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order for the original variable u in a broken H¹ norm and the auxiliary variable in norm are deduced without the boundedness of the numerical solution in ‐norm. Conversely, a linearized Crank‐Nicolson fully discrete scheme with the unconditional super close property is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.     

A two‐grid stabilized mixed finite element method for Darcy‐Forchheimer model

A two‐grid stabilized mixed finite element method based on pressure projection stabilization is proposed for the two‐dimensional Darcy‐Forchheimer model. We use the derivative of a smooth function, , to approximate the derivative of in constructing the two‐grid algorithm. The two‐grid method consists of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. There are a substantial reduction in computational cost. We prove the existence and uniqueness of solution of the discrete schemes on the coarse grid and the fine grid and obtain error estimates for the two‐grid algorithm. Finally, some numerical experiments are carried out to verify the accuracy and efficiency of the method.     

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

In this work, we study finite difference scheme for coupled time fractional Klein‐Gordon‐Schrödinger (KGS) equation. We proposed a linearized finite difference scheme to solve the coupled system, in which the fractional derivatives are approximated by some recently established discretization formulas. These formulas approximate the solution with second‐order accuracy at points different form the grid points in time direction. Taking advantage of this property, our proposed linearized scheme evaluates the nonlinear terms on the previous time level. As a result, iterative method is dispensable. The coupled terms in the scheme bring difficulties in analysis. By carefully studying these effects, we proved that the proposed scheme is unconditionally convergent and stable in discrete norm with energy method. Numerical results are included to justify the theoretical statements.     

Fourth‐order compact and energy conservative scheme for solving nonlinear Klein‐Gordon equation

In this article, a fourth‐order compact and conservative scheme is proposed for solving the nonlinear Klein‐Gordon equation. The equation is discretized using the integral method with variational limit in space and the multidimensional extended Runge‐Kutta‐Nyström (ERKN) method in time. The conservation law of the space semidiscrete energy is proved. The proposed scheme is stable in the discrete maximum norm with respect to the initial value. The optimal convergent rate is obtained at the order of in the discrete ‐norm. Numerical results show that the integral method with variational limit gives an efficient fourth‐order compact scheme and has smaller error, higher convergence order and better energy conservation for solving the nonlinear Klein‐Gordon equation compared with other methods under the same condition. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1283–1304, 2017     

High‐order difference scheme for the solution of linear time fractional klein–gordon equations

In this article, we apply a high‐order difference scheme for the solution of some time fractional partial differential equations (PDEs). The time fractional Cattaneo equation and the linear time fractional Klein–Gordon and dissipative Klein–Gordon equations will be investigated. The time fractional derivative which has been described in the Caputo's sense is approximated by a scheme of order , and the space derivative is discretized with a fourth‐order compact procedure. We will prove the solvability of the proposed method by coefficient matrix property and the unconditional stability and ‐convergence with the energy method. Numerical examples demonstrate the theoretical results and the high accuracy of the proposed scheme. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1234–1253, 2014     

New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

In this article, some high‐order accurate difference schemes of dispersive shallow water waves with Rosenau‐KdV‐RLW‐equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.     

Long‐time stability and asymptotic analysis of the IFE method for the multilayer porous wall model

In this article, we study the long‐time stability and asymptotic behavior of the immersed finite element (IFE) method for the multilayer porous wall model for the drug‐eluting stents. First, with the IFE method for the spatial descretization, and the implicit Euler scheme for the temporal discretization, respectively, we deduce the global stability of fully discrete solution. Then, we investigate the asymptotic behavior of the discrete scheme which reveals that the multilayer porous wall model converges to the corresponding elliptic equation if approaches to a steady‐state in both and norms as . Finally, some numerical experiments are given to verify the theoretical predictions.     

High‐order implicit staggered‐grid finite differences methods for the acoustic wave equation

Motivated by the idea that staggered‐grid methods give a greater stability and give energy conservation, this article presents a new family of high‐order implicit staggered‐grid finite difference methods with any order of accuracy to approximate partial differential equations involving second‐order derivatives. In particular, we numerically analyze our new methods for the solution of the one‐dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ‐point operator: the explicit formula is th‐order and the implicit formula is th‐order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.