The design of conservative finite element discretisations for the vectorial modified KdV equation

TL;DR

This paper introduces a Galerkin finite element scheme for the vectorial modified KdV equation that conserves energy precisely, enabling highly accurate long-term simulations and capturing complex soliton interactions.

Contribution

The paper presents a novel energy-conserving finite element discretisation for the vectorial modified KdV equation, ensuring minimal numerical dissipation and accurate long-term behavior.

Findings

Scheme conserves energy up to machine precision

Demonstrates asymptotic convergence with respect to discretisation

Successfully captures complex soliton interactions

Abstract

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.