A compatible finite element discretisation for the nonhydrostatic vertical slice equations

TL;DR

This paper introduces a compatible finite element discretisation for the vertical slice compressible Euler equations, employing an implicit time-stepping scheme with a scalable monolithic GMRES solver suitable for parallel computation.

Contribution

It presents a novel finite element discretisation at next-to-lowest order with an efficient, parallelizable solver for the vertical slice equations.

Findings

Robustness demonstrated on standard test cases

Solver allows large timesteps with linear iteration scaling

Implementation compatible with Firedrake and PETSc

Abstract

We present a compatible finite element discretisation for the vertical slice compressible Euler equations, at next-to-lowest order (i.e., the pressure space is bilinear discontinuous functions). The equations are numerically integrated in time using a fully implicit timestepping scheme which is solved using monolithic GMRES preconditioned by a linesmoother. The linesmoother only involves local operations and is thus suitable for domain decomposition in parallel. It allows for arbitrarily large timesteps but with iteration counts scaling linearly with Courant number in the limit of large Courant number. This solver approach is implemented using Firedrake, and the additive Schwarz preconditioner framework of PETSc. We demonstrate the robustness of the scheme using a standard set of testcases that may be compared with other approaches.