Overcoming order reduction in diffusion-reaction splitting. Part 2: oblique boundary conditions

TL;DR

This paper introduces a modification to classic splitting methods for diffusion-reaction equations with oblique boundary conditions, effectively overcoming order reduction and enhancing accuracy.

Contribution

A new correction to the Strang splitting method is proposed, resolving order reduction issues for problems with oblique boundary conditions.

Findings

Modified splitting schemes achieve higher order accuracy.

Numerical experiments confirm theoretical convergence rates.

Framework explains fractional convergence orders in classic methods.

Abstract

Splitting methods constitute a well-established class of numerical schemes for the time integration of partial differential equations. Their main advantages over more traditional schemes are computational efficiency and superior geometric properties. In the presence of non-trivial boundary conditions, however, splitting methods usually suffer from order reduction and some additional loss of accuracy. For diffusion-reaction equations with inhomogeneous oblique boundary conditions, a modification of the classic second-order Strang splitting is proposed that successfully resolves the problem of order reduction. The same correction also improves the accuracy of the classic first-order Lie splitting. The proposed modification only depends on the available boundary data and, in the case of non Dirichlet boundary conditions, on the computed numerical solution. Consequently, this modification can be implemented in an efficient way, which makes the modified splitting schemes superior to their classic versions. The framework employed in our error analysis also allows us to explain the fractional orders of convergence that are often encountered for classic Strang splitting. Numerical experiments that illustrate the theory are provided.