Irksome: Automating Runge-Kutta Time-stepping for Finite Element Methods

While implicit Runge-Kutta (RK) 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 Irksome, a high-level library for manipulating UM. (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.

Automated summary

Although implicit Runge-Kutta methods are accurate and stable, the implementation difficulties and high cost of solving the algebraic systems at each time step can be obstacles. Irksome is a high-level library that manipulates UM expressions to obtain UFL expressions for the coupled Runge-Kutta stage equations. It works with Firedrake package to efficiently solve the resulting algebraic systems, and numerical examples confirm its efficacy for various problems.