Discrete maximum-norm stability of a linearized second-order finite difference scheme for Allen–Cahn equation

In this paper, we use finite difference methods for solving the Allen–Cahn equation that contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three-level scheme in time and a second-order finite difference approach in space, and establish discrete boundedness stability in maximum norm: if the initial data are bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. It is shown that the main result can be obtained under certain restrictions on the time step.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

Pointwise error estimates of a linearized difference scheme for strongly coupled fractional Ginzburg-Landau equations

In this paper, a linearized semi-implicit finite difference scheme is proposed to solve the strongly coupled fractional Ginzburg-Landau equations. The difference scheme, which involves three time levels, is unconditionally stable, fourth-order accurate in space, and second-order accurate in time. By using the energy method and mathematical induction, the unique solvability, the unconditional stability, and optimal pointwise error estimate are obtained. Finally, some numerical experiments are presented to validate our theoretical findings.     

The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation

In the present paper, we analyze a second-order in time fully discrete finite element method for the BBM equation. The discretization in space is based on the standard Galerkin method, for the time discretization the Crank–Nicolson scheme is used. We also prove the convergence of a linearized Galerkin modification scheme.     

Unconditionally Optimal Error Analysis of the Second-Order BDF Finite Element Method for the Kuramoto-Tsuzuki Equation

This paper aims to study a second-order semi-implicit BDF finite element scheme for the Kuramoto-Tsuzuki equations in two dimensional and three dimensional spaces. The proposed scheme is stable and the nonlinear term is linearized by the extrapolation technique. Moreover, we prove that the error estimate in $L^2$-norm is unconditionally optimal which means that there has not any restriction on the time step and the mesh size. Finally, numerical results are displayed to illustrate our theoretical analysis.     

Quasi-acoustic scheme for the Euler equations of gas dynamics

We suggest an original scheme and an algorithm for the numerical solution of the Euler equations of gas dynamics. The construction of the scheme is based on the mass, momentum, and energy conservation laws. The flux computation is carried out by summation of elementary fluxes formed by small-amplitude running waves that satisfy the linearized equations of gas dynamics. The scheme contains no artificial regularizers, has second-order accuracy on smooth solutions, and is quasimonotone in a neighborhood of the discontinuities. Examples of one- and two-dimensional computations are given.     

An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

In this paper, we propose a linearized implicit finite difference scheme for solving the fractional Ginzburg-Landau equation. The scheme, which involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. Moreover, the unique solvability, the unconditional stability, and the convergence of the method in the \(L^{\infty }\)-norm are proved by the energy method and mathematical induction. Compared with the implicit midpoint difference scheme (Wang and Huang J. Comput. Phys. 312, 31–49, 2016), current linearized method generally reduces the computational cost. Finally, numerical results are presented to confirm the theoretical results.     

Discrete shock profiles for systems of conservation laws

The existence of discrete shock profiles for difference schemes approximating a system of conservation laws is the major topic studied in this paper. The basic theorem established here applies to first-order accurate difference schemes; for weak shocks, this theorem provides necessary and sufficient conditions involving the truncation error of the linearized scheme which guarantee entropy satisfying or entropy violating discrete shock profiles. Several explicit difference schemes are used as examples illustrating the interplay between the entropy condition, monotonicity, and linearized stability. Entropy violating stationary shocks for second-order accurate Lax-Wendroff schemes approximating systems are also constructed. The only tools used in the proofs are local analysis and the center manifold theorem.     

On Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation

In this paper, we have proved the second-order convergence in L_∞ norm of the Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation. The existence, uniqueness and iterative algorithm are also discussed in detail. Furthermore, a L_∞ second order convergent linearized difference scheme is given for inhomogeneous equation. All results are obtained without any restrictions on the meshsizes. At last a numerical example is presented.     

一类组合KdV方程的数值方法

1引言从Scott-Russell[1]提出在平静的水面上孤波运动的情形以后,大量的目光开始关注它的存在性、其中的性质和动态的交互情形[2],这是因为许多非线性动态的物理现象可以被描述成一个孤立子的模型[2,3].诸如,Korteweg-de Vries(KdV)方程[4,5,6],正弦     

An efficient nonlinear iteration scheme for nonlinear parabolic-hyperbolic system

A nonlinear iteration method named the Picard-Newton iteration is studied for a two-dimensional nonlinear coupled parabolic-hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization-discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard-Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard-Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard-Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.     

高阶广义KDV方程和KP方程的数值解法

采用一种线性隐格式来解3-阶和5-阶的广义Korteweg-De Vries(KDV)方程,对这种方法做一推广,就能应用到它的二维形式广义Kadomtsew-Petviashvili(KP)方程．这种方法是无条件稳定的,且是无损耗的．数值实验描述了一个线性孤波运动的情形以及两个孤波交互的情形,从结果来看,它们满足孤立子解的两个守恒-动量守恒和能量守恒．     

A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time.     

An efficient nonlinear iteration scheme for nonlinear parabolic–hyperbolic system

A nonlinear iteration method named the Picard–Newton iteration is studied for a two-dimensional nonlinear coupled parabolic–hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization–discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard–Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard–Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard–Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.     

守恒型扩散方程非线性离散格式的性质分析和快速求解

对于守恒型扩散方程,研究其二阶时间精度非线性全隐有限差分离散格式的性质,证明了其解的存在唯一性.研究了二阶时间精度的Picard-Newton迭代格式,证明了迭代解对原问题真解的二阶时间和空间收敛性,以及对非线性离散解的二次收敛速度,实现了非线性问题的快速求解.本文中方法也适用于一阶时间精度格式的分析,并可推广至对流扩散问题.数值实验验证了二阶时间精度Picard-Newton迭代格式的高精度和高效率.     

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

A numerical algorithm for simulation of the Q-switched fiber laser using the travelling wave model

We develop a finite-difference scheme for approximation of a system of nonlinear PDEs describing the Q-switching process. We construct it by using staggered grids. The transport equations are approximated along characteristics, and quadratic nonlinear functions are linearized using a special selection of staggered grids. The stability analysis proves that a connection between time and space steps arises only due to approximation requirements in order to follow exactly the directions of characteristics. The convergence analysis of this scheme is done in two steps. First, some estimates of the uniform boundedness of the discrete solution are proved. This part of the analysis is done locally, in some neighborhood of the exact solution. Second, on the basis of the obtained estimates, the main stability inequality is proved. The second-order convergence rate with respect to the space and time coordinates follows from this stability estimate. Using the obtained convergence result, we prove that the local stability analysis in the selected neighborhood of the exact solution is sufficient.     

A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schrödinger equations

In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE). The scheme has the property of mass and energy conservation at the discrete level, with an unconditional stability and a second-order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.     

Fast algorithm for viscous Cahn-Hilliard equation

The main purpose of this paper is to solve the viscous Cahn-Hilliard equation via a fast algorithm based on the two time-mesh (TT-M) finite element (FE) method to ease the problem caused by strong nonlinearities. The TT-M FE algorithm includes the following main computing steps. First, a nonlinear FE method is applied on a coarse time-mesh τ_c. Here, the FE method is used for spatial discretization and the implicit second-order θ scheme (containing both implicit Crank-Nicolson and second-order backward difference) is used for temporal discretization. Second, based on the chosen initial iterative value, a linearized FE system on time fine mesh is solved, where some useful coarse numerical solutions are found by Lagrange’s interpolation formula. The analysis for both stability and a priori error estimates is made in detail. Numerical examples are given to demonstrate the validity of the proposed algorithm. Our algorithm is compared with the traditional Galerkin FE method and it is evident that our fast algorithm can save computational time.     

An L2 finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.