Stage-Parallel Implicit Runge--Kutta methods via low-rank matrix equation corrections

TL;DR

This paper introduces a parallel approach for implicit Runge--Kutta methods that decouples stage computations using low-rank matrix corrections, significantly improving efficiency for large-scale stiff ODEs.

Contribution

It presents a novel low-rank correction technique to decouple IRK stage equations, enabling efficient parallel computation and extending to nonlinear problems.

Findings

Achieves stable decoupling of stage equations via low-rank corrections.

Demonstrates improved efficiency in numerical experiments with PDE discretizations.

Provides shared memory parallel implementation and analysis.

Abstract

Implicit Runge--Kutta (IRK) methods are highly effective for solving stiff ordinary differential equations (ODEs) but can be computationally expensive for large-scale problems due to the need of solving coupled algebraic equations at each step. This study improves IRK efficiency by leveraging parallelism to decouple stage computations and reduce communication overhead, specifically we stably decouple a perturbed version of the stage system of equations and recover the exact solution by solving a Sylvester matrix equation with an explicitly known low-rank right-hand side. Two IRK families -- symmetric methods and collocation methods -- are analyzed, with extensions to nonlinear problems using a simplified Newton method. Implementation details, shared memory parallel code, and numerical examples, particularly for ODEs from spatially discretized PDEs, demonstrate the efficiency of the proposed IRK technique.