# On time relaxed schemes and formulations for dispersive wave equations

## TL;DR

This paper introduces relaxed system formulations for dispersive wave equations that simplify high order derivatives, enabling stable and efficient numerical simulations of nonlinear water waves using standard methods.

## Contribution

It proposes new relaxed formulations that approximate dispersive wave equations with only first order derivatives, improving numerical stability and simplicity.

## Key findings

- Successfully applied to the Korteweg-de Vries equation
- Reduces stability restrictions on time steps
- Enables use of standard numerical methods

## Abstract

The numerical simulation of nonlinear dispersive waves is a central research topic of many investigations in the nonlinear wave community. Simple and robust solvers are needed for numerical studies of water waves as well. The main difficulties arise in the numerical approximation of high order derivatives and in severe stability restrictions on the time step, when explicit schemes are used. In this study we propose new relaxed system formulations which approximate the initial dispersive wave equation. However, the resulting relaxed system involves first order derivatives only and it is written in the form of an evolution problem. Thus, many standard methods can be applied to solve the relaxed problem numerically. In this article we illustrate the application of the new relaxed scheme on the classical Korteweg-de Vries equation as a prototype of stiff dispersive PDEs.