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For simulations of time evolution problems, such as weather and climatemodels, taking the largest stable timestep is advantageous for reducingwall-clock time. A drawback of doing so is the potential reduction in nonlinearaccuracy of the numerical solution - we investigate this for the RotatingShallow Water Equations (RSWEs) on an f-plane. First, we examine how lineardispersion errors can impact the nonlinear dynamics. By deriving an alternatetime evolution equation for the RSWEs, the dynamics can be expressed throughinteractions of three linear waves in triads. Linear dispersion errors mayappear in the numerical representation of the frequency of each triad, whichwill impact the timestepped nonlinear dynamics. A new triadic error quantifiesthis by composing three stability polynomials from the oscillatory Dahlquisttest equation. Second, we design two new test cases to examine the effect oftimestep size in a numerical model. These tests investigate how well atimestepper replicates slow nonlinear dynamics amidst fast linear oscillations.The first test case of a Gaussian height perturbation contains a nonlinearphase shift that can be missed with a large timestep. The second set of triadictest cases excite a few linear waves to instigate specific triadicinteractions. Two triadic cases are examined: one with a dominant directlyresonant triad and another with near-resonances that redistribute fast modeenergy into rings in spectral space. Three numerical models, including LFRicfrom the Met Office, are examined in these test cases with differenttimesteppers.

The effect of linear dispersive errors on nonlinear timestepping accuracy in the f-plane rotating shallow water equations