  For simulations of time-evolution problems, such as weather and climate
models, taking the largest stable time-step is advantageous for reducing the
wall-clock time. We propose methods for studying the effect of linear
dispersive errors on the time-stepping accuracy of nonlinear problems. We
demonstrate an application of this to the Rotating Shallow Water Equations
(RSWEs). To begin, a nonlinear time-stepping `triadic error' metric is
constructed from three-wave interactions. Stability polynomials, obtained from
the oscillatory Dahlquist test equation, enable the computation of triadic
errors for different time-steppers; we compare five classical schemes. We next
provide test cases comparing different time-step sizes within a numerical
model. The first case is of a reforming Gaussian height perturbation. This
contains a nonlinear phase shift that can be missed with a large time-step. The
second set of test cases initialise individual waves to allow specific triads
to form. The presence of a slow transition from linear to nonlinear dynamics
creates a good venue for testing how the slow phase information is replicated
with a large time-step. Three models, including the finite element code Gusto,
and the MetOffice's new LFRic model, are examined in these test cases with
different time-steppers.
