A semi‐implicit, semi‐Lagrangian discontinuous Galerkin framework for adaptive numerical weather prediction

We present an adaptive discretization approach for model equations typical of numerical weather prediction (NWP), which combines the semi‐Lagrangian technique with a semi‐implicit time discretization method, based on the Trapezoidal Rule second‐order Backward Difference Formula scheme (TR‐BDF2), and with a discontinuous Galerkin (DG) spatial discretization, with variable and adaptive element degree. The resulting method has full second‐order accuracy in time, can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element at runtime, in order to balance accuracy and computational cost. Furthermore, although the proposed method can be implemented on arbitrary unstructured and non‐conforming meshes, even its application on simple Cartesian meshes in spherical coordinates can reduce the impact of the coordinate singularity, by reducing the polynomial degree used in the polar elements. Numerical results are presented, obtained on classical benchmarks with two‐dimensional models implementing discretizations of the shallow‐water equations on the sphere and of the Euler equations on a vertical slice, respectively. The results confirm that the proposed method has a significant potential for NWP applications.