Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.Comment: accepted versio

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

This is the final version. Available from Elsevier via the DOI in this record.We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor–Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We demonstrate that our discretisation achieves the expected second order convergence and provide results from some standard rotating sphere test problems.Natural Environment Research Council (NERC)Natural Environment Research Council (NERC)Engineering and Physical Sciences Research Council (EPSRC)Engineering and Physical Sciences Research Council (EPSRC

A Domain-Specific Language and Editor for Parallel Particle Methods

Domain-specific languages (DSLs) are of increasing importance in scientific high-performance computing to reduce development costs, raise the level of abstraction and, thus, ease scientific programming. However, designing and implementing DSLs is not an easy task, as it requires knowledge of the application domain and experience in language engineering and compilers. Consequently, many DSLs follow a weak approach using macros or text generators, which lack many of the features that make a DSL a comfortable for programmers. Some of these features---e.g., syntax highlighting, type inference, error reporting, and code completion---are easily provided by language workbenches, which combine language engineering techniques and tools in a common ecosystem. In this paper, we present the Parallel Particle-Mesh Environment (PPME), a DSL and development environment for numerical simulations based on particle methods and hybrid particle-mesh methods. PPME uses the meta programming system (MPS), a projectional language workbench. PPME is the successor of the Parallel Particle-Mesh Language (PPML), a Fortran-based DSL that used conventional implementation strategies. We analyze and compare both languages and demonstrate how the programmer's experience can be improved using static analyses and projectional editing. Furthermore, we present an explicit domain model for particle abstractions and the first formal type system for particle methods.Comment: Submitted to ACM Transactions on Mathematical Software on Dec. 25, 201

Code generation for generally mapped finite elements

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations

Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction

We present an energy conserving space discretisation based on a Poisson bracket that can be used to derive the dry compressible Euler as well as thermal shallow water equations. It is formulated using the compatible finite element method, and extends the incorporation of upwinding for the shallow water equations as described in Wimmer, Cotter, and Bauer (2019). While the former is restricted to DG upwinding, an energy conserving SUPG scheme for the (partially) continuous Galerkin thermal field space is newly introduced here. The energy conserving property is validated by coupling the Poisson bracket based spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved temperature field development with respect to stability when upwinding is included. An approximately energy conserving full discretisation with a smaller computational cost is also presented.Comment: 27 pages, 9 figures, first version: all comments welcom