Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations

Abstract

This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the scheme numerically for the rotating shallow water equations. In particular, we investigate whether the scheme conserves stationary solutions, represents well the nonlinear dynamics, and approximates well the frequency relations of the continuous equations, while preserving conservation laws such as mass and total energy.

Variational integrator for the rotating shallow‐water equations on the sphere

“…see [3] and §A.3 for details. It is this expression of the RSW equations that appears in a discretized form in the variational discretization later in (23).…”

“…Numerical integrators that discretely preserve one or more of the aforementioned geometric properties are presented e.g. in [3,4,5,7,8,10,17,23,24,25,27,32].…”

“…In recent years, considerable effort was devoted to the development of structure-preserving integrators on general structured and unstructured grids, see e.g. [3,6,12,15,26,28,29].…”

“…While most of the work on variational integration was devoted to ordinary differential equations, recent years have seen an increased interest in the partial differential equation case, see e.g. [25] and [3,4,11] for some applications of the variational methodology to important models of geophysical fluid dynamics. Variational integrators for the partial differential equations of fluid dynamics are designed by replacing the continuous configuration space of the model equation, represented as an appropriate infinite-dimensional Lie group, by a suitable finite dimensional matrix Lie group on which the variational principle can be applied in both its Lagrangian and Eulerian versions, thanks to an application of the Euler-Poincaré reduction theorem.…”

A structure-preserving approximation of the discrete split rotating shallow water equations

“…The construction of such schemes is an active area of research and various approaches to develop structure-preserving discretizations exist: e.g. variational discretizations [5,6,14] or compatible FE methods [9,11]. In particular FE methods are a very general, widely applicable approach allowing for flexible use of meshes and higher order approximations.…”