Nonlinear Compact Finite-Difference Schemes with Semi-Implicit Time Stepping

Atmospheric flows are characterized by a large range of length scales as well as strong gradients. The accurate simulation of such flows requires numerical algorithms with high spectral resolution, as well as the ability to provide nonoscillatory solutions across regions of high gradients. These flows exhibit a large range of time scales as well—the slowest waves propagate at the flow velocity and the fastest waves propagate at the speed of sound. Time integration with explicit methods are thus inefficient, although algorithms with semi-implicit time integration have been used successfully in past studies. We propose a finite-difference method for atmospheric flows that uses a weighted compact scheme for spatial discretization and implicit-explicit additive Runge-Kutta methods for time integration. We present results for a benchmark atmospheric flow problem and compare our results with existing ones in the literature.