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

TL;DR

This paper demonstrates that the Strang splitting method achieves second-order convergence for parabolic PDEs with oblique boundary conditions when combined with the Crank-Nicolson scheme, even with nonlinearities, unlike other schemes.

Contribution

It proves superconvergence of Strang splitting with Crank-Nicolson for certain boundary conditions, highlighting order reduction issues with other schemes.

Findings

Strang splitting with Crank-Nicolson is second-order accurate for the problem.

Order reduction occurs with other Runge-Kutta schemes or exact flows.

Numerical experiments confirm second-order convergence with nonlinearities.

Abstract

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 prove these results when the source term only depends on the space variable, an assumption which makes the splitting scheme equivalent to the Crank-Nicolson method itself applied to the whole problem. Numerical experiments suggest that the second order convergence persists with general nonlinearities.