Quantifying the accuracy of large time-steps in highly oscillatory systems

There is an ever-increasing demand for longer and higher resolution numerical simulations of the weather and climate. To achieve this with reasonable wall-clock times, it is desirable to use as large a time-step as possible, whilst retaining a stable and accurate solution. However, this is very challenging in the presence of highly oscillatory linear waves; there are explicit time-step limits and losses in accuracy with implicit methods. This talk will highlight where non-linear errors can result from large time-steps and provide metrics for quantifying this. We begin by re-casting the non-linearity as a product of linear waves. In the Rotating Shallow Water Equations (RSWEs), this allows for key dynamics to be expressed as three-wave ‘triad’ interactions. A non-linear ‘triadic’ time-stepping error is computed using linear stability polynomials. A number of explicit and implicit time-stepping methods (such as RK4, TR-BDF2, ETD-RK2) will be compared analytically in the RSWEs. Next, two new test problems enable analyses of large time-step simulations. The first is of a Gaussian perturbation to a RSWE height field. A proposed metric, relating to the kinetic energy distribution over temporal frequency, quantifies phase errors in the height reformation. The second test case initialises linear waves which, via direct- and near- resonant triad interactions, will construct non-linear dynamics. Phase errors with large time-steps can be identified in the corresponding height fields. A first variant of this case will initialise only two waves; this will primarily instigate an energy exchange within a dominant triad. A second version, containing more slow modes, enables a re-distribution of fast mode energy into rings in wavenumber space.