Improved accuracy for time-splitting methods for the numerical solution of parabolic equations

TL;DR

This paper introduces enhanced time-splitting domain decomposition methods for parabolic equations that significantly reduce splitting errors, enabling more accurate and parallelizable numerical solutions compared to classical schemes.

Contribution

The authors develop a novel correction technique that improves splitting accuracy from second to third order, extending ADI methods with parallelizable algorithms for reaction-diffusion problems.

Findings

Splitting error reduced to $ ext{O}( au^3)$ with correction.

New schemes outperform classical ADI and Crank-Nicolson methods.

Parallel subdomain solutions improve computational efficiency.

Abstract

In this work, we study time-splitting strategies for the numerical approximation of evolutionary reaction-diffusion problems. In particular, we formulate a family of domain decomposition splitting methods that overcomes some typical limitations of classical alternating direction implicit (ADI) schemes. The splitting error associated with such methods is observed to be $\mathcal{O}(\tau^2)$ in the time step $\tau$. In order to decrease the size of this splitting error to $\mathcal{O}(\tau^3)$, we add a correction term to the right-hand side of the original formulation. This procedure is based on the improved initialization technique proposed by Douglas and Kim in the framework of ADI methods. The resulting non-iterative schemes reduce the global system to a collection of uncoupled subdomain problems that can be solved in parallel. Computational results comparing the newly derived algorithms with the Crank-Nicolson scheme and certain ADI methods are presented.