  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.
