Operator splitting for well-posed active scalar equations

TL;DR

This paper studies the convergence of operator splitting methods, specifically Godunov and Strang, applied to well-posed active scalar equations with nonlinear advection and various linear operators, ensuring reliable numerical solutions.

Contribution

It provides a rigorous analysis of the convergence rates of splitting methods for a broad class of active scalar equations, including fractional diffusion and dispersive equations.

Findings

Godunov and Strang splitting methods converge with expected rates

Convergence holds for equations like surface quasi-geostrophic and aggregation equations

Results apply to equations with fractional diffusion and dispersive terms

Abstract

We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator, and a linear (local or nonlocal) diffusion operator or a linear dispersion operator. The advection velocity is determined from the scalar unknown itself and hence the equations are so-called active scalar equations. Examples are provided by the surface quasi-geostrophic and aggregation equations. In addition, Burgers-type equations with fractional diffusion as well as the KdV and Kawahara equations are covered. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data is sufficiently regular.