Jacobian-free implicit MDRK methods for stiff systems of ODEs

TL;DR

This paper introduces a family of implicit multiderivative Runge-Kutta methods for stiff ODEs, analyzing two derivative computation strategies and demonstrating the benefits of separate formulas despite larger Newton systems.

Contribution

It proposes a novel multistage MDRK family based on approximate implicit Taylor methods, comparing derivative computation approaches and analyzing their conditioning effects.

Findings

Direct derivative computation leads to exponential conditioning increase with stiffness.

Separate formulas for derivatives result in linear conditioning dependence on stiffness.

Numerical results show benefits of separate formulas despite larger Newton systems.

Abstract

In this work, an approximate family of implicit multiderivative Runge-Kutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza et al. in Comput. Appl. Math. 39:304, 2020). As a Taylor method can be written in MDRK format, the novel family constitutes a multistage generalization. Two different alternatives are investigated for the computation of the higher order derivatives: either directly as part of the stage equation, or either as a separate formula for each derivative added on top of the stage equation itself. From linearizing through Newton's method, it turns out that the conditioning of the Newton matrix behaves significantly different for both cases. We show that direct computation results in a matrix with a conditioning that is highly dependent on the stiffness, increasing exponentially in the stiffness parameter with the amount of derivatives. Adding separate formulas has a more favorable behavior, the matrix conditioning being linearly dependent on the stiffness, regardless of the amount of derivatives. Despite increasing the Newton system significantly in size, through several numerical results it is demonstrated that doing so can be considerably beneficial.