A new approach to constructing efficient stiffly accurate exponential propagation iterative methods of Runge-Kutta type (EPIRK)

TL;DR

This paper develops new stiff order conditions and algorithms for EPIRK exponential integrators, enhancing their efficiency and stability for stiff problems, especially when combined with adaptive Krylov methods.

Contribution

It extends the theoretical framework for EPIRK methods by deriving stiff order conditions and introduces a novel construction approach optimized for adaptive Krylov algorithms.

Findings

New EPIRK schemes demonstrate improved efficiency on stiff problems.

Constructed methods outperform previous exponential integrators in numerical tests.

Enhanced stability and accuracy for stiff differential equations.

Abstract

The structural flexibility of the exponential propagation iterative methods of Runge-Kutta type (EPIRK) enables construction of particularly efficient exponential time integrators. While the EPIRK methods have been shown to perform well on stiff problems, all of the schemes proposed up to now have been derived using classical order conditions. In this paper we extend the stiff order conditions and the convergence theory developed for the exponential Rosenbrock methods to the EPIRK integrators. We derive stiff order conditions for the EPIRK methods and develop algorithms to solve them to obtain specific schemes. Moreover, we propose a new approach to constructing particularly efficient EPIRK integrators that are optimized to work with an adaptive Krylov algorithm. We use a set of numerical examples to illustrate the computational advantages that the newly constructed EPIRK methods offer compared to previously proposed exponential integrators.