Convergence of a numerical scheme for a coupled Schr\"odinger--KdV system

TL;DR

This paper proves the convergence of a finite difference numerical scheme for a coupled Schr"odinger--KdV system, which models wave interactions, and demonstrates its effectiveness through numerical examples.

Contribution

It introduces a new convergence proof for a semi-discrete scheme for the coupled Schr"odinger--KdV system, overcoming challenges due to energy estimate limitations.

Findings

Convergence of the numerical scheme is established in a strong norm.

Numerical examples confirm the theoretical convergence results.

Abstract

We prove the convergence in a strong norm of a finite difference semi-discrete scheme approximating a coupled Schr\"odinger--KdV system on a bounded domain. This system models the interaction of short and long waves. Since the energy estimates available in the continuous case do not carry over to the discrete setting, we rely on a suitably truncated problem which we prove reduces to the original one. We present some numerical examples to illustrate our convergence result.