Operator splitting for well-posed active scalar equations
Provides theoretical convergence guarantees for operator splitting methods in a broad class of nonlinear PDEs, benefiting numerical analysts and computational scientists working on these equations.
The paper proves convergence with expected rates for Godunov and Strang splitting methods applied to active scalar equations (e.g., surface quasi-geostrophic, aggregation, Burgers-type with fractional diffusion, KdV, Kawahara) under sufficient regularity of initial data.
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.