A semi-discrete numerical scheme for nonlocally regularized KdV-type equations

TL;DR

This paper introduces a semi-discrete numerical scheme for a broad class of nonlocally regularized KdV-type equations, proving convergence and analyzing localization errors, with numerical experiments validating the theoretical results.

Contribution

The authors develop and analyze a semi-discrete numerical method for nonlocal KdV-type equations, including convergence proof and error estimates for localization.

Findings

Numerical scheme converges uniformly as mesh size decreases.

Localization error diminishes with larger finite domains.

Numerical experiments confirm theoretical error bounds.

Abstract

A general class of KdV-type wave equations regularized with a convolution-type nonlocality in space is considered. The class differs from the class of the nonlinear nonlocal unidirectional wave equations previously studied by the addition of a linear convolution term involving third-order derivative. To solve the Cauchy problem we propose a semi-discrete numerical method based on a uniform spatial discretization, that is an extension of a previously published work of the present authors. We prove uniform convergence of the numerical method as the mesh size goes to zero. We also prove that the localization error resulting from localization to a finite domain is significantly less than a given threshold if the finite domain is large enough. To illustrate the theoretical results, some numerical experiments are carried out for the Rosenau-KdV equation, the Rosenau-BBM-KdV equation and a convolution-type integro-differential equation. The experiments conducted for three particular choices of the kernel function confirm the error estimates that we provide.