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Finite differences and Runge-Kutta time stepping schemes used inComputational AeroAcoustics simulations are often optimized for low dispersionand dissipation (e.g. DRP or LDDRK schemes) when applied to linear problems inorder to accurately simulate waves with the least computational cost. Here, theperformance of optimized Runge-Kutta time stepping schemes for lineartime-invariant problems with non-constant-amplitude oscillations is considered.This is in part motivated by the recent suggestion that optimized spatialderivatives perform poorly for growing and decaying waves, as theiroptimization implicitly assumes real wavenumbers. To our knowledge, this is thefirst time the time-stepping of non-constant-amplitude oscillations has beenconsidered. It is found that current optimized Runge-Kutta schemes performpoorly in comparison with their maximal order equivalents fornon-constant-amplitude oscillations. Moreover, significantly more accurateresults can be achieved for the same computation cost by replacing a two-stepscheme such as LDDRK56 with a single step higher-order scheme with a longertime step. Attempts are made at finding optimized schemes that perform well fornon-constant-amplitude oscillations, and three such examples are provided.However, the traditional maximal order Runge-Kutta time stepping schemes arestill found to be preferable for general problems with broadband excitation.These theoretical predictions are illustrated using a realistic 1Dwave-propagation example.

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