A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation

To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation.

UNIFORM DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED LINEAR 2ND ORDER HYPERBOLIC PROBLEM WITH ZEROTH ORDER REDUCED EQUATION

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work.It has the good property of discrete conservative energy.By the discrete energy analysis and mathematical induction method,it is proved to be uniquely solvable and unconditionally convergent with the secondorder accuracy in both time and space.Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law,unique solvability and unconditional convergence of order two in time and four in space.The resultant schemes preserve the maximum bound principle.The analysis techniques for convergence used in this paper also work for the Euler scheme,the Crank-Nicolson scheme and others.Numerical experiments are carried out to verify the computational efficiency,conservative law and the maximum bound principle of the proposed difference schemes.

Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov （KGZ） equation. The new scheme is also decoupled in computation, which means that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in imple- mentation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L∞- norm, and for N in the discrete L2-norm, respectively, where U and N are the numeri- cal solutions of the KGZ equation. Numerical results verify the theoretical analysis.

On The Numerical Solution of Two Dimensional Model of an Alloy Solidification Problem

In this paper, a linearized three level difference scheme is derived for two-dimensional model of an alloy solidification problem called Sivashinsky equation. Further, it is proved that the scheme is uniquely solvable and convergent with convergence rate of order two in a discrete L∞-norm. At last, numerical experiments are carried out to support the theoretical claims.

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR QUASILINEAR PARABOLIC DIFFERENTIAL EQUATION

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L2 norm is proved and numerical examples are presented.