Abstract
A multiple integral finite volume method combined and Lagrange interpolation are applied in this paper to the Rosenau-RLW (RRLW) equation. We construct a two-level implicit fully discrete scheme for the RRLW equation. The numerical scheme has the accuracy of third order in space and second order in time, respectively. The solvability and uniqueness of the numerical solution are shown. We verify that the numerical scheme keeps the original equation characteristic of energy conservation. It is proved that the numerical scheme is convergent in the order of O(tau ^{2} + h^{3}) and unconditionally stable. A numerical experiment is given to demonstrate the validity and accuracy of scheme.
Highlights
It is well known that nonlinear partial differential equations exist in many areas of mathematical physics and fluid mechanics
The RLW equation was proposed by Peregrine [1, 2] based on the classical Korteweg– de Vries (KdV) equation, and an explanation of different situations of a nonlinear dispersive wave was given in his research
In Ref. [26], the Galerkin cubic B-spline finite element method is proposed to construct the numerical scheme for the RRLW equation
Summary
It is well known that nonlinear partial differential equations exist in many areas of mathematical physics and fluid mechanics. The coupling equation of KdV and RRLW is solved through a three-level average implicit finite difference scheme, showing second-order accuracy in space and time, simultaneously [25]. [26], the Galerkin cubic B-spline finite element method is proposed to construct the numerical scheme for the RRLW equation. Pan [27] investigated the C-N scheme of RRLW equation through a more classic finite difference approach, and corresponding solvability and convergence have been proved. The main contribution of the current work is to present a two-level implicit numerical scheme for the following RRLW equation with some theoretic analysis: ut – uxxt + uxxxxt + ux + uux = 0, (x, t) ∈ (xl, xr) × [0, T]. We discuss the discrete energy conservative laws of the numerical scheme and prove its solvability and uniqueness in Sect. We explain the concrete and detailed methods about improving the accuracy of the numerical discrete scheme
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