Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions

TL;DR

This paper introduces a new splitting method for diffusion-reaction equations with Dirichlet boundary conditions that avoids order reduction, ensuring higher accuracy and positivity preservation in numerical simulations.

Contribution

A novel splitting procedure that maintains second-order accuracy and positivity preservation despite complex boundary conditions, supported by rigorous convergence analysis.

Findings

The new method avoids order reduction with inhomogeneous Dirichlet conditions.

Numerical simulations confirm higher accuracy in 1D and 2D cases.

Convergence analysis validates the method's reliability.

Abstract

For diffusion-reaction equations employing a splitting procedure is attractive as it reduces the computational demand and facilitates a parallel implementation. Moreover, it opens up the possibility to construct second-order integrators that preserve positivity. However, for boundary conditions that are neither periodic nor of homogeneous Dirichlet type order reduction limits its usefulness. In the situation described the Strang splitting procedure is not more accurate than Lie splitting. In this paper, we propose a splitting procedure that, while retaining all the favorable properties of the original method, does not suffer from order reduction. We demonstrate our results by conducting numerical simulations in one and two space dimensions with inhomogeneous and time dependent Dirichlet boundary conditions. In addition, a mathematical rigorous convergence analysis is conducted that confirms the results observed in the numerical simulations.