Irksome: Automating Runge--Kutta time-stepping for finite element methods

TL;DR

Irksome is a high-level library that automates the implementation of implicit Runge--Kutta methods for finite element problems, simplifying the solution of coupled algebraic systems and improving computational efficiency.

Contribution

The paper introduces Irksome, a novel library that automates Runge--Kutta time-stepping in finite element methods, integrating with Firedrake for efficient algebraic system solutions.

Findings

Irksome effectively automates Runge--Kutta implementation.

Numerical examples demonstrate improved efficiency and accuracy.

The library simplifies complex algebraic system solutions.

Abstract

While implicit Runge--Kutta methods possess high order accuracy and important stability properties, implementation difficulties and the high expense of solving the coupled algebraic system at each time step are frequently cited as impediments. We present IIrksome, a high-level library for manipulating UFL (Unified Form Language) expressions of semidiscrete variational forms to obtain UFL expressions for the coupled Runge--Kutta stage equations at each time step. Irksome works with the Firedrake package to enable the efficient solution of the resulting coupled algebraic systems. Numerical examples confirm the efficacy of the software and our solver techniques for various problems.