Extending Irksome: improvements in automated Runge--Kutta time stepping for finite element methods

TL;DR

This paper discusses recent enhancements to the Irksome library, enabling more efficient and flexible automated Runge--Kutta time-stepping methods for finite element PDE discretizations, including improved solvers and preconditioners.

Contribution

The paper introduces new formulations of Runge--Kutta methods, optimized support for DIRK methods, and advanced preconditioning tools, advancing the automation and efficiency of time-stepping in finite element PDE solvers.

Findings

Enhanced solver performance demonstrated through computational examples.

Effective preconditioning strategies for implicit Runge--Kutta systems.

Support for diagonally implicit methods improves computational efficiency.

Abstract

Irksome is a library based on the Unified Form Language (UFL) that enables automated generation of Runge--Kutta methods for time-stepping finite element spatial discretizations of partial differential equations (PDE). Allowing users to express semidiscrete forms of PDE, it generates UFL representations for the stage-coupled variational problems to be solved at each time step. The Firedrake package then generates efficient code for evaluating these variational problems and allows users a wide range of options to deploy efficient algebraic solvers in PETSc. In this paper, we describe several recent advances in Irksome. These include alternate formulations of the Runge--Kutta time-stepping methods and optimized support for diagonally implicit (DIRK) methods. Additionally, we present new and improved tools for building preconditioners for the resulting linear and linearized systems, demonstrating that these can lead to efficient approaches for solving fully implicit Runge-Kutta discretizations. The new features are demonstrated through a sequence of computational examples demonstrating the high-level interface and obtained solver performance.