Convergence of numerical schemes for short wave long wave interaction equations

TL;DR

This paper proves the convergence of semi-discrete finite volume numerical schemes for a coupled nonlinear Schrödinger and hyperbolic conservation law system modeling short and long wave interactions, supported by numerical examples.

Contribution

It establishes the convergence of finite volume schemes for a coupled PDE system using compensated compactness, a novel approach for this interaction model.

Findings

Convergence of numerical solutions to the entropy solution.

Validation through numerical experiments.

Application of compensated compactness method.

Abstract

We consider the numerical approximation of a system of partial differential equations involving a nonlinear Schr\"odinger equation coupled with a hyperbolic conservation law. This system arises in models for the interaction of short and long waves. Using the compensated compactness method, we prove convergence of approximate solutions generated by semi-discrete finite volume type methods towards the unique entropy solution of the Cauchy problem. Some numerical examples are presented.