Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation

TL;DR

This paper introduces a novel symbolic-numeric finite difference approach for the modified KdV equation that ensures the preservation of local conservation laws of mass and energy, enhancing numerical stability and physical fidelity.

Contribution

It develops a new class of finite difference methods that exactly preserve two local conservation laws of the mKdV equation using a symbolic-numeric approach.

Findings

Methods preserve mass and energy conservation laws.

Enhanced numerical stability demonstrated.

Applicable to other nonlinear PDEs with conservation laws.

Abstract

By exploiting the fact that conservation laws form the kernel of a discrete Euler operator, we use a recently introduced symbolic-numeric approach to construct a new class of finite difference methods for the modified Korteweg-de Vries (mKdV) equation, that preserve the local conservation laws of mass and energy.