Optimized Runge-Kutta (LDDRK) timestepping schemes for non-constant-amplitude oscillations

TL;DR

This paper evaluates the performance of optimized Runge-Kutta schemes for non-constant-amplitude oscillations in wave simulations, revealing limitations of current schemes and proposing improved alternatives for specific cases.

Contribution

It is the first study to analyze Runge-Kutta time-stepping for non-constant-amplitude oscillations and compares their effectiveness to maximal order schemes.

Findings

Current optimized schemes perform poorly for non-constant-amplitude oscillations.

Single higher-order schemes with longer time steps can outperform two-step schemes.

Traditional maximal order schemes are preferable for broadband excitation problems.

Abstract

Finite differences and Runge-Kutta time stepping schemes used in Computational AeroAcoustics simulations are often optimized for low dispersion and dissipation (e.g. DRP or LDDRK schemes) when applied to linear problems in order to accurately simulate waves with the least computational cost. Here, the performance of optimized Runge-Kutta time stepping schemes for linear time-invariant problems with non-constant-amplitude oscillations is considered. This is in part motivated by the recent suggestion that optimized spatial derivatives perform poorly for growing and decaying waves, as their optimization implicitly assumes real wavenumbers. To our knowledge, this is the first time the time-stepping of non-constant-amplitude oscillations has been considered. It is found that current optimized Runge-Kutta schemes perform poorly in comparison with their maximal order equivalents for non-constant-amplitude oscillations. Moreover, significantly more accurate results can be achieved for the same computation cost by replacing a two-step scheme such as LDDRK56 with a single step higher-order scheme with a longer time step. Attempts are made at finding optimized schemes that perform well for non-constant-amplitude oscillations, and three such examples are provided. However, the traditional maximal order Runge-Kutta time stepping schemes are still found to be preferable for general problems with broadband excitation. These theoretical predictions are illustrated using a realistic 1D wave-propagation example.