Operator splitting for well-posed active scalar equations

2012/01/30 by Helge Holden, Kenneth H. Karlsen, Holden, Helge +3
Engineering · Mathematics · #35Q35 #65M12 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1201.6254

openalex publication_date 2012/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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