Semi-Lagrangian exponential integration for geophysical flows

Weather and climate models must handle both highly oscillatory linear dynamics and important nonlinear advection effects while maintaining computational efficiency. To achieve large time-step sizes, models often use semi-Lagrangian advection methods combined with semi-implicit treatments of oscillatory linear dynamics. However, this can lead to dispersion effects, such as the attenuation of wave speed in fast atmospheric waves. Exponential integrators are numerical methods that can accurately solve highly oscillatory linear dynamics, allowing for large time steps and potential added parallelism. However, when these integrators are used with nonlinear dynamics, they typically require smaller time steps. In this talk, we present a class of Semi-Lagrangian Exponential schemes. These schemes handle the exponential of the linear operator within a Lagrangian framework at each time step, aiming to bypass the time-step limitations of traditional exponential schemes and improve the solution of linear oscillatory dynamics compared to classic semi-implicit methods. The framework was first proposed in Peixoto & Schreiber (2019) and applied to the shallow water equations on an f-plane, demonstrating the feasibility of large time steps with exponential schemes. Recent work by Steinstraesser, Peixoto, and Schreiber (2024) has enhanced the accuracy and stability of these schemes, applying them to the rotating shallow water equations on the sphere. This new approach leverages the strengths of exponential integrators for linear dynamics and semi-Lagrangian methods for nonlinear advection, promising more efficient numerical simulations of geophysical fluid dynamics.