Operator splitting for partial differential equations with Burgers nonlinearity

TL;DR

This paper introduces a new analytical approach to operator splitting methods for nonlinear PDEs like Burgers and KdV, demonstrating convergence rates and applicability to various equations with well-posedness.

Contribution

It provides a novel analytical framework for operator splitting in nonlinear PDEs, establishing convergence rates for equations including Burgers, KdV, and others.

Findings

Strang splitting converges at the expected rate for these equations.

Second-order convergence in H^r for KdV with initial data in H^{r+5}.

Applicable to a range of nonlinear PDEs with well-posedness.

Abstract

We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+u u_x$ where $A$ is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers' equation, the Korteweg-de Vries (KdV) equation, the Benney-Lin equation, and the Kawahara equation. We show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular. In particular, for the KdV equation we obtain second-order convergence in $H^r$ for initial data in $H^{r+5}$ with arbitrary $r\ge 1$.