Analysis of a Seamless Semi-Implicit Finite Volume Method for Atmospheric Flows

In this contribution, we present a non-standard, functional analytic framework to rigorously analyze the properties of the semi-implicit second-order finite volume discretization for the compressible Euler equations developed by Benacchio and Klein (2019). In experiments, this method shows favorable stability properties for geophysically relevant benchmarks and is capable of approximating the pseudo-incompressible and/or hydrostatic limit regime without changing the underlying discretization.
Our main results are the consistency and stability of the implicit projection step, which we achieve by choosing discontinuous velocity and continuous pressure variables. As a consequence, the classical divergence is replaced by its natural analogue, i.e., by line integrals along the boundary of a dual cell.
In contrast to preceding work, we consider general quadrilateral and cuboid meshes, and we provide an interpolation operator that is compatible with the natural divergence on the dual grid.
Aiming for a rigorous stability estimate of the overall scheme, we furthermore discuss a choice of advection operator that ensures compatibility.