Traveling-wave solutions and structure-preserving numerical methods for a hyperbolic approximation of the Korteweg-de Vries equation

TL;DR

This paper investigates a hyperbolic approximation of the Korteweg-de Vries equation, analyzing its solutions and developing structure-preserving numerical methods that are energy-conserving and effective in simulations.

Contribution

It introduces a hyperbolic model of KdV with diverse solutions and develops novel energy-preserving, structure-preserving numerical discretizations for this model.

Findings

The hyperbolic KdV approximation captures soliton and periodic solutions.

The proposed numerical schemes are asymptotic preserving and energy-conserving.

Numerical experiments confirm the effectiveness of the discretizations.

Abstract

We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate KdV solitons, as well as other solitary and periodic solutions that are related to higher-order water wave models, and may include singularities. We analyze a class of implicit-explicit Runge-Kutta time discretizations for KdVH that are asymptotic preserving, energy conserving, and can be applied to other hyperbolized systems. We also develop structure-preserving spatial discretizations based on summation-by-parts operators in space including finite difference, discontinuous Galerkin, and Fourier methods. We use the entropy relaxation approach to make the fully discrete schemes energy-preserving. Numerical experiments demonstrate the effectiveness of these discretizations.