Rosenbrock-Wanner Time Integration in 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 convergence 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. Rather than simply using four iterations of a second order Crank-Nicolson time discretisation as is customary, the present article studies the impact of using various third-order, four stage Rosenbrock-Wanner schemes, where instead of a simple time centering, the integration weights are chosen to meet specific stability and order conditions. 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, a desirable property in the presence of fast wave dynamics across multiple scales. These schemes are compared to four iterations of a Crank-Nicolson scheme for the solution of the 2D rotating shallow water equations at the 3D compressible Euler equations at both planetary and non-hydrostatic scales are are shown to exhibit improved results in terms of their energetic profiles and stability.