High order operator splitting methods based on an integral deferred correction framework

TL;DR

This paper introduces high order operator splitting schemes based on an integral deferred correction framework, which improve accuracy and stability in solving initial value problems and PDEs, demonstrated through numerical experiments.

Contribution

It develops a novel IDC-based framework for high order operator splitting methods, enhancing accuracy by correcting splitting and numerical errors.

Findings

Achieves arbitrary high order accuracy with IDC corrections.

Demonstrates improved stability and efficiency in solving PDEs.

Validates the approach with numerical examples in two dimensions.

Abstract

Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the IDC procedure to solve initial value problems (IVPs). We present analysis to show that the IDC methods can correct for both the splitting and numerical errors, lifting the order of accuracy by $r$ with each correction, where $r$ is the order of accuracy of the method used to solve the correction equation. We further apply this framework to solve partial differential equations (PDEs). Numerical examples in two dimensions of linear and nonlinear initial-boundary value problems are presented to demonstrate the performance of the proposed IDC approach.