Abstract
A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and second-order convergent. Numerical examples show the efficiency of the scheme.
Highlights
In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW equation: ut − uxxt uxxxxt ux up x 0 xl < x < xr, 0 < t < T, 1.1 with an initial condition u x, 0 u0 x xl ≤ x ≤ xr, 1.2 and boundary conditions u xl, t u xr, t 0, uxx xl, t uxx xr, t 0 0 ≤ t ≤ T, Boundary Value Problems where p ≥ 2 is a integer and u0 x is a known smooth function
In 2 Li and Vu-Quoc said “. . . in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation”
In 3–11, some conservative finite difference schemes were used for a system of the generalized nonlinear Schrodinger equations, Regularized long wave RLW equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively
Summary
We consider the following initial-boundary value problem of the general Rosenau-RLW equation: ut − uxxt uxxxxt ux up x 0 xl < x < xr , 0 < t < T , 1.1 with an initial condition u x, 0 u0 x xl ≤ x ≤ xr , 1.2 and boundary conditions u xl, t u xr , t 0, uxx xl, t uxx xr , t 0 0 ≤ t ≤ T , Boundary Value Problems where p ≥ 2 is a integer and u0 x is a known smooth function. When p 2, 1.1 is called as usual Rosenau-RLW equation. When p 3, 1.1 is called as modified Rosenau-RLW MRosenau-RLW equation. The initial boundary value problem 1.1 – 1.3 possesses the following conservative quantities: Qt xr u x, t dx 2 xl
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