Stable corrections for perturbed diagonally implicit Runge--Kutta methods

TL;DR

This paper develops stable correction techniques for perturbed diagonally implicit Runge--Kutta methods, enabling efficient low-accuracy computations while maintaining stability and accuracy in numerical solutions.

Contribution

It introduces novel stabilized correction approaches that improve the stability of mixed accuracy DIRK methods with low-accuracy implicit solves.

Findings

Stability of mixed accuracy DIRK methods is thoroughly analyzed.

New stabilized correction techniques are proposed and validated.

The methods achieve better accuracy and stability for larger time-steps.

Abstract

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.