Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with oblique boundary conditions

Guillaume Bertoli , Christophe Besse (1) , Gilles Vilmart

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We also show that this method recovers stationary states in contrast with splitting methods in general. We prove these results when the source term only depends on the space variable. Numerical experiments suggest that the second order convergence persists with general nonlinearities.

Mots clés: Strang splitting; Crank-Nicolson; diffusion-reaction equation; nonhomogeneous boundary conditions; order reduction

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Guillaume Bertoli, Christophe Besse, Gilles Vilmart. Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with oblique boundary conditions. 2020. ⟨hal-02992821v1⟩