Additive Polynomial Time Integrators, Part I: Framework and Fully-Implicit-Explicit (FIMEX) Collocation Methods

TL;DR

This paper introduces a flexible polynomial time integration framework for additive problems, focusing on new implicit-explicit methods that improve stability and efficiency, especially for PDEs and parallel computations.

Contribution

It develops a general additive integration framework and introduces high-order, stable FIMEX collocation methods based on Radau nodes with enhanced computational efficiency.

Findings

FIMEX integrators show improved stability over existing IMEX methods.

Parallelization significantly speeds up computations for PDEs with trivial implicit parts.

New methods achieve higher accuracy and faster runtimes in finite-element discretizations.

Abstract

In this paper we generalize the polynomial time integration framework to additively partitioned initial value problems. The framework we present is general and enables the construction of many new families of additive integrators with arbitrary order-of-accuracy and varying degree of implicitness. In this first work, we focus on a new class of implicit-explicit polynomial block methods that are based on fully-implicit Runge-Kutta methods with Radau nodes, and possess high stage order. We show that the new fully-implicit-explicit (FIMEX) integrators have improved stability compared to existing IMEX Runge-Kutta methods, while also being more computationally efficient due to recent developments in preconditioning techniques for solving the associated systems of nonlinear equations. For PDEs on periodic domains where the implicit component is trivial to invert, we will show how parallelization of the right-hand-side evaluations can be exploited to obtain significant speedup compared to existing serial IMEX Runge-Kutta methods. For parallel (in space) finite-element discretizations, the new methods can achieve orders of magnitude better accuracy than existing IMEX Runge-Kutta methods, and/or achieve a given accuracy several times times faster in terms of computational runtime.