A Comparison of Rosenbrock-Wanner and Crank-Nicolson Time Integrators for Atmospheric Modelling

Non-hydrostatic atmospheric models often use semi-implicit temporal discretisations in order to negate the time step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the non-linear solver is typically truncated to a fixed number of iterations, using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third-order, four stage Rosenbrock-Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank-Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock-Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly-stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow water equations and the 3D compressible Euler equations at both planetary and non-hydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank-Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock-Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at non-hydrostatic scales.