Operator splitting for the KdV equation

TL;DR

This paper introduces a new analytical approach to operator splitting methods for equations like the KdV, demonstrating convergence under certain regularity conditions for initial data.

Contribution

It provides a novel analytical framework for operator splitting, specifically for equations with linear and quadratic components, including the KdV equation.

Findings

Godunov and Strang splitting methods converge with expected rates

Convergence depends on initial data regularity

Applicable to equations of the form u_t=Au+B(u)

Abstract

We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+B(u)$ where $A$ is a linear operator and $B$ is quadratic. A particular example is the Korteweg-de Vries (KdV) equation $u_t-u u_x+u_{xxx}=0$. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.