Automated Runge-Kutta-Nystr\"om time stepping for finite element methods in Irksome

TL;DR

This paper introduces enhancements to the Irksome library enabling automated implementation of Runge-Kutta-Nystr"om methods for second-order PDEs, improving efficiency and accuracy in finite element simulations.

Contribution

It extends Irksome to handle second-order temporal derivatives directly with Runge-Kutta-Nystr"om methods, reducing algebraic system size and improving computational performance.

Findings

Runge-Kutta-Nystr"om methods outperform standard methods for wave equations with proper preconditioning.

The approach is effective for high-order spatial derivatives and complex systems like poroelasticity.

Numerical results show competitive performance with explicit methods and robustness near incompressible limits.

Abstract

Irksome is a library based on the Unified Form Language (UFL) that automates the application of Runge-Kutta time-stepping methods for finite element spatial discretizations of partial differential equations (PDEs). This paper describes recent updates to Irksome that allow users to express semidiscrete forms of PDEs that contain second-order temporal derivatives, whence it generates stage-coupled variational problems to be solved at each time step for Runge-Kutta-Nystr\"om methods. Firedrake then generates code for these variational problems and provides a rich interface to PETSc for solving them. Directly discretizing second-order time derivatives with Runge-Kutta-Nystr\"om methods provides several advantages relative to discretizing a rewritten first-order system with a standard Runge-Kutta method. Besides working with an interface closer to the problem formulation in UFL, avoiding these auxiliary variables means that Runge-Kutta-Nystr\"om methods lead to smaller algebraic systems and better run-time. Our numerical results indicate that, with effective preconditioning, fully implicit Runge-Kutta-Nystr\"om methods can be made competitive with more traditional explicit methods for wave equations. They are also (essentially) required to discretize wave-type equations with higher-order spatial derivatives. We also provide numerical experiments for fully dynamic poroelasticity, a system of mixed temporal order, where our time-stepping and algebraic solvers perform effectively even as we approach the incompressible limit.