462 research outputs found

    Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

    Full text link
    We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

    Compatible finite element methods for geophysical fluid dynamics

    Get PDF
    This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretising the transport terms that arise in dynamical core equation systems, and provide some example discretisations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretisations, Poisson bracket discretisations, and consistent vorticity transport.Comment: correction of some typo

    A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids

    Full text link
    A novel wetting and drying treatment for second-order Runge-Kutta discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water equations is proposed. It is developed for general conforming two-dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a non-destructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well-balancing with an innovative velocity-based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme -- even on unstructured grids -- and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near-realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, super-linear convergence, mass-conservation, well-balancedness, and stability are verified

    A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies

    Full text link
    We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only first order derivatives, thus allowing to overcome the numerical difficulties and the severe time step restrictions arising from higher order terms. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an energy conservation law from which the energy conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation recently proposed in "C. Escalante, M. Dumbser and M.J. Castro. An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes, Journal of Computational Physics 2018"

    A Quasi-Hamiltonian Discretization of the Thermal Shallow Water Equations

    Get PDF
    International audienceThe rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model

    Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations -- I: numerical scheme and validation

    Full text link
    We present an energy/entropy stable and high order accurate finite difference method for solving the linear/nonlinear shallow water equations (SWE) in vector invariant form using the newly developed dual-pairing (DP) and dispersion-relation preserving (DRP) summation by parts (SBP) finite difference operators. We derive new well-posed boundary conditions for the SWE in one space dimension, formulated in terms of fluxes and applicable to linear and nonlinear problems. For nonlinear problems, entropy stability ensures the boundedness of numerical solutions, however, it does not guarantee convergence. Adequate amount of numerical dissipation is necessary to control high frequency errors which could ruin numerical simulations. Using the dual-pairing SBP framework, we derive high order accurate and nonlinear hyper-viscosity operator which dissipates entropy and enstrophy. The hyper-viscosity operator effectively tames oscillations from shocks and discontinuities, and eliminates poisonous high frequency grid-scale errors. The numerical method is most suitable for the simulations of sub-critical flows typical observed in atmospheric and geostrophic flow problems. We prove a priori error estimates for the semi-discrete approximations of both linear and nonlinear SWE. We verify convergence, accuracy and well-balanced property via the method of manufactured solutions (MMS) and canonical test problems such as the dam break, lake at rest, and a two-dimensional rotating and merging vortex problem.Comment: 32 pages, 10 figures, comments are welcom

    A structure-preserving split finite element discretization of the split wave equations

    Get PDF
    We introduce a new finite element (FE) discretization framework applicable for covariant split equations. The introduction of additional differential forms (DF) that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric-dependent closure equations that apply the Hodge-star operator. Our discretization framework conserves this geometrical structure and provides for all DFs proper FE spaces such that the differential operators hold in strong form. We introduce lowest possible order discretizations of the split 1D wave equations, in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant FE spaces, omitting partial integrations. Approximating the Hodge-star by nontrivial Galerkin projections (GP), the two discrete metric equations follow by projections onto either the piecewise constant (GP0) or piecewise linear (GP1) space. Our framework gives us three schemes with significantly different behavior. The split scheme using twice GP1 is unstable and shares the dispersion relation with the P1-P1 FE scheme that approximates both variables by piecewise linear spaces (P1). The split schemes that apply a mixture of GP1 and GP0 share the dispersion relation with the stable P1-P0 FE scheme that applies piecewise linear and piecewise constant (P0) spaces. However, the split schemes exhibit second order convergence for both quantities of interest. For the split scheme applying twice GP0, we are not aware of a corresponding standard formulation to compare with. Though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast, the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find FE schemes by new combinations of FE spaces
    corecore