Boosting the Accuracy of Finite Difference Schemes via Optimal Time Step Selection and Non-Iterative Defect Correction

TL;DR

This paper introduces a simple, efficient method to enhance the accuracy of finite difference schemes for time-dependent PDEs by optimally choosing the time step and applying non-iterative defect correction, significantly improving results with minimal extra effort.

Contribution

The paper proposes a novel combination of optimal time step selection and non-iterative defect correction to boost finite difference scheme accuracy without extensive modifications.

Findings

Enhanced accuracy of finite difference schemes demonstrated across multiple PDE types.

Method reduces computational cost while increasing solution precision.

Applicable to various spatial dimensions and domain geometries.

Abstract

In this article, we present a simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non-iterative manner. The power of the technique is its ability to extract as much accuracy as possible from existing finite difference schemes with minimal additional effort. Through straightforward numerical analysis arguments, we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method. We demonstrate the utility of optimal time step (OTS) selection combined with non-iterative defect correction (NIDC) on several different types of finite difference schemes for a wide array of classical linear and semilinear PDEs in one and more space dimensions on both regular and irregular domains.