462 research outputs found
Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems
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
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
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
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
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
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
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
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