European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS22: Structure-preserving discretization methods II: Variational methods for physics compatible discretizations (Part 1)
Chair: Marc Gerritsma
25 mins
A structure-preserving approximation of the discrete split rotating shallow water equations
Werner Bauer, Jörn Behrens, Colin Cotter
Abstract: We introduce an efficient split finite element (FE) discretization of a y-independent (slice) model of the rotating shallow water equations. The study of this slice model provides insight towards developing schemes for the full 2D case. Using the split Hamiltonian FE framework (Bauer, Behrens and Cotter, 2019), we result in structure-preserving discretizations that are split into topological prognostic and metric-dependent closure equations. This splitting also accounts for the schemes' properties: the Poisson bracket is responsible for conserving energy (Hamiltonian) as well as mass, potential vorticity and enstrophy (Casimirs), independently from the realizations of the metric closure equations. The latter, in turn, determine accuracy, stability, convergence and discrete dispersion properties. We exploit this splitting to introduce structure-preserving approximations of the mass matrices in the metric equations avoiding to solve linear systems. We obtain a fully structure-preserving scheme with increased efficiency by a factor of two.
25 mins
Structure-Preserving Global Ocean Modelling on Unstructured Grids
Peter Korn
Abstract: We describe the structure-preserving discretization of the ocean model ICON-O, the ocean component of Max-Planck Institute for Meteorology's newly developed Earth System Model ICON-ESM and the ocean model of the ICON modelling system. ICON-O is a global general circulation model, formulated on triangular cells with an Arakawa C-type staggering. The models PDE’s as well as its oceanic SGS use a coherent structure-preserving discretization that incorporates ideas from finite-element, finite volume and mimetic discretization methods. We present numerical and experimental analysis of global ocean simulation to demonstrate the physical quality of the model solution and the computational efficiency of the model towards the goal of establishing unstructured grid models as a viable and attractive alternative to established structured grid models. An outlook presents ongoing work towards a structure-preserving extension of our numerical scheme.
25 mins
Structure-preserving models of reversible and irreversible dynamics in geophysical fluids
Christopher Eldred, Thomas Dubos, Francois Gay-Balmaz
Abstract: The reversible (entropy-conserving) dynamics of geophysical fluids (and fluids more generally) have a well-known geometric structure in terms of a Hamiltonian formulation, consisting of a Hamiltonian and Poisson brackets. A powerful tool for the construction of numerical models of geophysical fluids with many desirable properties is then to discretize the Poisson bracket and the Hamiltonian themselves, in a way that preserves the essential parts of their structure. A general approach to do this is to use a mimetic spatial discretization combined with an energy-conserving Poisson time integrator, giving what are known as structure-preserving models. These models have many useful properties, including discrete conservation of total mass, energy and entropy. Here we use mimetic Galerkin differences on block-structured grids (MGD, a type of tensor-product compatible Galerkin method) coupled with a second-order, implicit energy-conserving Poisson integrator based on the average vector field discrete gradient. The MGD element avoids spectral gaps and other dispersive anomalies found with more standard compatible finite element methods. However, real geophysical fluids also have irreversible (entropy-generating) processes, such as phase changes, momentum/thermal dissipation and diffusion, which conserve total energy and mass; and generate entropy, in accordance with the first and second laws of thermodynamics. In this case, the (less well known) geometric structure is a metriplectic formulation, which combines a Poisson bracket for the reversible dynamics with a metric (or dissipation) bracket for the irreversible dynamics. These processes usually occur below the grid scale and are therefore parameterized in terms of the resolved, reversible dynamics. This talk will present a discretization of this geometric structure in the context of the nonhydrostatic fully compressible Euler equations with some typical subgrid turbulence parameterizations, using the same spatial and temporal discretizations. In doing so, for the first time, to machine precision, the reversible dynamics (still) exactly conserve total mass, energy and entropy; and the irreversible dynamics (parameterizations) conserve total mass and entropy and generate entropy. Results using this model from planar versions of the commonly used DCMIP test suite will be shown. If time permits, there will be some discussion of future work on the extension of these ideas to multicomponent/multiphase fluids (including moisture), more sophisticated turbulence parameterizations and other areas of geophysical fluids/physics more generally.
25 mins
Solution of the 3D compressible Euler equations using mixed mimetic spectral elements
David Lee, Artur Palha
Abstract: This talk will discuss the development of a solver for the 3D compressible Euler equations for atmospheric flows on a rotating sphere using a mixed mimetic spectral element discretisation. Particular care is taken to preserve the skew-symmetric structure of the coupled system in the discrete form, so as to satisfy the exact balance of kinetic, potential and internal energy exchanges. The formulation of a preconditioner for the acceleration of implicit time stepping methods will also be discussed. The model is validated against a standard test case for baroclinic instability within an otherwise geostrophically and hydrostatically balanced atmosphere.