# Locally conservative finite difference schemes for the modified KdV equation

## TL;DR

This paper develops new finite difference schemes for the modified KdV equation that preserve key conservation laws, using a symbolic approach extended to cubic nonlinearities, resulting in highly accurate, mass-preserving numerical methods.

## Contribution

It extends a symbolic conservation-preserving scheme approach to cubic nonlinear PDEs, producing new second-order accurate, mass-preserving finite difference schemes for the modified KdV equation.

## Key findings

- Several new families of schemes preserving mass and energy or momentum.
- Some schemes are highly accurate compared to existing methods.
- Includes Average Vector Field schemes adapted for the modified KdV.

## Abstract

Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity. In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and coworkers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.