On the convergence of a high-accuracy compact conservative scheme for the modified regularized long-wave equation

Abstract

In this article, we develop a high-order efficient numerical scheme to solve the initial-boundary problem of the MRLW equation. The method is based on a combination between the requirement to have a discrete counterpart of the conservation of the physical “energy” of the system and finite difference method. The scheme consists of a fourth-order compact finite difference approximation in space and a version of the leap-frog scheme in time. The unique solvability of numerical solutions is shown. A priori estimate and fourth-order convergence of the finite difference approximate solution are discussed by using discrete energy method and some techniques of matrix theory. Numerical results are given to show the validity and the accuracy of the proposed method.

Değiştirilmiş düzenli uzun dalga denkleminin kübik trigonometrik B-spline fonksiyonları kullanılarak nümerik çözümü

“…where the parameters , = 1, … ,7 are determined from system (14), in which these values are not prefered to document here due to being too long.…”

“…Thus, the equation has been often taken up by numerical analysts and different solution methods have been developed. The finite difference [1,2], the homotopy perturbation [3], the Adomian decomposition [4], the expilicit multisymplectic [5], the He's variational iteration [6], the meshless [7], the homotopy anlysis [8], the Galerkin linear finite element [9], the second-order Fourier pseudospectral [10], the explicit multistep Galerkin finite element [11], the mixed Galerkin finite element [12], the split least-squares mixed finite element [13], the compact conservative [14] and the moving least square collocation [15] methods are some of those methods. In addition to those ones, there are also some methods that are obtained by using various B-spline functions.…”