A new conservative nonlinear high-order compact finite difference scheme for the general Rosenau-RLW equation

Huan Wang,Jue Wang,Shuguang Li

doi:10.1186/s13661-015-0336-2

Huan Wang, Jue Wang + Show 1 more

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https://doi.org/10.1186/s13661-015-0336-2

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Journal: Boundary Value Problems	Publication Date: May 8, 2015
Citations: 5	License type: CC BY 4.0

Affiliation: Harbin Engineering University

Abstract

In this paper, a new conservative high-order compact finite difference scheme is studied for the initial-boundary value problem of the generalized Rosenau-regularized long wave equation. We design new conservative nonlinear fourth-order compact finite difference schemes. It is proved by the discrete energy method that the compact scheme is uniquely solvable; we have the energy conservation and the mass conservation for this approach in discrete Sobolev spaces. The convergence and stability of the difference schemes are obtained, and its numerical convergence order is $O(\tau^{2}+h^{4})$ in the $L^{\infty}$ -norm. Furthermore, numerical results are given to support the theoretical analysis. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.

Highlights

In this paper, we consider the following initial-boundary value problem of the Generalized Rosenau-RLW Regularized Long Wave (RLW) equation (GRRLW): ut + uxxxxt – uxxt + ux + up x =, (x, t) ∈ × (, T], ( . )u(xl, t) = u(xr, t) =, uxx(xl, t) = uxx(xr, t) =, t ∈ (, T], u(x, ) = u (x), x ∈, where p ≥ is a positive integer, = and u (x) are known smooth functions
We introduce the discrete L -inner product and the associated norm
A uxx(xi) ≈ δx Ui ⇒ uxx(xi) ≈ A– δx Ui, A ux(xi) ≈ δx Ui ⇒ ux(xi) ≈ A– δx Ui, A uxxxx(xi) ≈ δx Ui ⇒ uxxxx(xi) ≈ A– δx Ui, where Ui is the approximation of u(xi)

Summary

Introduction

We consider the following initial-boundary value problem of the Generalized Rosenau-RLW Regularized Long Wave (RLW) equation (GRRLW): ut + uxxxxt – uxxt + ux + up x = , (x, t) ∈ × In [ – ], some new finite difference schemes for the initial-boundary value problem of the RLW equation were considered. In [ ], a new conservative difference scheme for the general Rosenau-RLW equation was proposed. In [ ], Pan and Zhang proposed a conservative linearized difference scheme for the general Rosenau-RLW equation which was unconditionally stable and second-order convergent and simulates conservative laws at the same time. In Section , the prior error estimates for a fourth-order finite difference approximation of the GRRLW equation are obtained, and the convergence and stability of the difference scheme are proved.

Vin h

Thus we have

Taking the inner product of with

Table shows the numerical results when τ