Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part

TL;DR

This paper develops super-convergent IMEX-Peer methods with A-stable implicit parts, enhancing efficiency and stability for solving differential equations with stiff and non-stiff components.

Contribution

It extends IMEX-Peer methods to a broader class, deriving conditions for super-convergence and optimizing methods for stability and accuracy.

Findings

Constructed super-convergent IMEX-Peer methods of orders 3, 4, 5.

Achieved favorable stability properties through a search algorithm.

Demonstrated improved performance via numerical experiments.

Abstract

In this paper, we extend the implicit-explicit (IMEX) methods of Peer type recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203--215, 2017] to a broader class of two-step methods that allow the construction of super-convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of explicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super-convergent IMEX-Peer methods with favourable stability properties, we derive necessary and sufficient conditions on the coefficient matrices and apply an extrapolation approach based on already computed stage values. Optimised super-convergent IMEX-Peer methods of order s+1 for s=2,3,4 stages are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other IMEX-Peer methods are included.