Multirate Infinitesimal Step Flux Splitting Methods In Numerical Weather Prediction

Numerical simulations of atmospheric models resolve slow (advection) and fast (gravity and sound) modes, creating significant challenges in handling multiple time scales under CFL restrictions. In this work, we develop time-space flux splitting schemes for the two-dimensional compressible Euler equations. We introduce flux splitting within the framework of the multirate infinitesimal step (MIS) method and formulate the proposed schemes using the discontinuous Galerkin flux-differencing approach to ensure compatibility with high-order spatial discretizations. We integrate the advective terms using a Runge-Kutta method with a macro time step constrained by the CFL condition, while we treat sound wave terms with smaller time steps to satisfy stability requirements dictated by the speed of sound. Additionally, the fast waves are resolved using a Rosenbrock-type method, where the Jacobian is constructed as one-dimensional operator in the vertical direction. We investigate different approaches, present an extension to curvilinear coordinates, analyze the stability properties of the new schemes, and present several atmospheric test cases, comparing the results to existing methods.