Smooth perturbations of diagonally implicit Runge--Kutta methods

TL;DR

This paper develops a framework for designing high-order diagonally implicit Runge--Kutta methods that use smooth perturbations to reduce computational cost while maintaining accuracy and stability.

Contribution

It introduces novel methods that replace the original operator with a lower accuracy one for smooth perturbations, ensuring high order and stability.

Findings

New high-order perturbed DIRK methods verified through numerical experiments.

Strategies to mitigate the impact of smooth perturbations on accuracy and stability.

Demonstrated computational savings using lower accuracy operators in implicit stages.

Abstract

A mixed accuracy framework for Runge--Kutta methods presented in [Grant, JSC 2022] has been shown to speed up the computation in diagonally implicit Runge--Kutta (DIRK) methods by using less expensive low accuracy approaches for the implicit stages. This theory included both smooth and nonsmooth perturbations, and subsequent work focused primarily on the case of nonsmooth perturbations that arise from mixed precision simulations. In this work the focus is on smooth perturbations that arise from using less accurate models or under-resolved iterative solvers to simplify the implicit computations. We develop an accuracy and stability analysis based on the framework in [Grant, JSC 2022] to design methods that strategically replace the original operator by a lower accuracy operator to reduce computational cost while mitigating the effect of the perturbations. In particular, we focus on designing novel methods that are high order for smooth perturbations that satisfy additional local consistency conditions. Finally, we verify the performance of the novel perturbed DIRK methods designed in this work and numerically study the impact of different types of smooth perturbations on the accuracy and stability of the methods.