Philipp Birken presenting "An entropy stable well-balanced high order low Mach solver for atmospheric flows"

Abstract. 

We consider numerical methods for the dynamical core in numerical weather prediction (NWP). Specifically, we are interested in schemes for the Euler equations in 3D at low Mach numbers with a gravity source term. As the trend in NWP is to increase the resolution, high order Discontinuous Galerkin (DG) methods are competitive, proving well suited for High Performance Computing due to their high arithmetic intensity. Due to stiffness, implicit discretizations are needed. The overall design goal is thus a suitable highly parallel stable and computational efficient DG implementation.

To deal with the gravitational source term, a well-balanced method is needed, suitable for large domains. Additionally, the scheme requires a numerical flux suitable for low Mach numbers, and some form of nonlinear stability such as entropy stability, suitable for large simulation times.

Environment and Climate Change Canada (ECCC) is currently redesigning its dynamical core along these lines within the WxFactory code. We present the implementation of a well-balanced DG method suitable for low Mach numbers. Specifically, we use a well-balanced Direct Flux Reconstruction implementation on GL nodes on quadrilateral meshes and a variant of the AUSM+up flux. For time integration, both implicit Runge-Kutta methods, exponential integrators and ImEx methods are being looked at. To achieve entropy stability, we are in the process of incorporating an artificial diffusion term and a relaxation technique to stabilize the time integration. We provide numerical results showing the performance on atmospheric test cases in 2D and 3D.

This is joint work with Stéphane Gaudreault, Shoyon Panday, Carlos Pereira Frontado (ECCC), and Herman Efraimsson, Martin Hindle, Kateryna Tymoshchuk (LU).