Timestepping stability in pseudospectral methods

The maximum numerically stable timestep size when explicitly timestepping problems with nonlinear advection is typically limited by the smallest spatial scales in the simulation. But applying this convention to pseudospectral simulations may be unnecessarily restrictive. This work presents simulations of a 1D Korteweg-De Vries equation and a 2D Rayleigh-Bénard convection system. Each problem consists of flow through boundaries where the problem fields are expanded in spectral bases with non-uniform collocation grids. As we take progressively larger timesteps, the first timestepping instabilities to occur are found to be driven by large-scale modes in the spectral decomposition, rather than being triggered by CFL-type violations in the smallest-scale modes. We find that such instabilities are consistent with an asymptotic analysis of the local errors of a given timestepping scheme. These results motivate the viability of taking large timesteps and illuminate the behavior of timestepping schemes in pseudospectral simulations.