A mass, energy, vorticity, and potential enstrophy conserving lateral boundary scheme for the shallow water equations using piecewise linear boundary approximations

2011 ◽  
Vol 230 (8) ◽  
pp. 2751-2793 ◽  
Author(s):  
G.S. Ketefian ◽  
M.Z. Jacobson
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Piecewise Linear ◽  
Lateral Boundary ◽  
Linear Boundary ◽  
Mass Energy ◽  
Potential Enstrophy

scholarly journals A Potential Enstrophy and Energy Conserving Scheme for the Shallow-Water Equations Extended to Generalized Curvilinear Coordinates

2017 ◽  
Vol 145 (3) ◽  
pp. 751-772 ◽  
Author(s):  
Michael D. Toy ◽  
Ramachandran D. Nair
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Empirical Test ◽  
Normal Mode Analysis ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
Mode Analysis ◽  
Convergence Error ◽  
Starting Point ◽  
Potential Enstrophy

An energy and potential enstrophy conserving finite-difference scheme for the shallow-water equations is derived in generalized curvilinear coordinates. This is an extension of a scheme formulated by Arakawa and Lamb for orthogonal coordinate systems. The starting point for the present scheme is the shallow-water equations cast in generalized curvilinear coordinates, and tensor analysis is used to derive the invariant conservation properties. Preliminary tests on a flat plane with doubly periodic boundary conditions are presented. The scheme is shown to possess similar order-of-convergence error characteristics using a nonorthogonal coordinate compared to Cartesian coordinates for a nonlinear test of flow over an isolated mountain. A linear normal mode analysis shows that the discrete form of the Coriolis term provides stationary geostrophically balanced modes for the nonorthogonal coordinate and no unphysical computational modes are introduced. The scheme uses centered differences and averages, which are formally second-order accurate. An empirical test with a steady geostrophically balanced flow shows that the convergence rate of the truncation errors of the discrete operators is second order. The next step will be to adapt the scheme for use on the cubed sphere, which will involve modification at the lateral boundaries of the cube faces.

scholarly journals Enriched Galerkin method for the shallow-water equations

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Moritz Hauck ◽  
Vadym Aizinger ◽  
Florian Frank ◽  
Hennes Hajduk ◽  
Andreas Rupp
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Degrees Of Freedom ◽  
Piecewise Linear ◽  
Lower Number ◽  
Approximation Spaces ◽  
Continuous Galerkin ◽  
Dg Method ◽  
Implementation Aspects ◽  
Locally Conservative

AbstractThis work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, element-wise constant functions. Similar to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallow-water equations is well-balanced, in the sense that it preserves lake-at-rest configurations. We evaluate the method’s robustness and accuracy using various analytical and realistic problems and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLAB / GNU Octave framework FESTUNG.

scholarly journals Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods – Part 1: Derivation and properties

2017 ◽  
Vol 10 (2) ◽  
pp. 791-810 ◽  
Author(s):  
Christopher Eldred ◽  
David Randall
Keyword(s):  
Total Energy ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Models ◽  
Standard Test ◽  
Exterior Calculus ◽  
Hamiltonian Methods ◽  
Discrete Analogues ◽  
Work Done ◽  
Potential Enstrophy

Abstract. The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.

scholarly journals Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods: Derivation and Properties (Part 1)

2016 ◽  
Author(s):  
Christopher Eldred ◽  
David Randall
Keyword(s):  
Total Energy ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Models ◽  
Standard Test ◽  
Exterior Calculus ◽  
Hamiltonian Methods ◽  
Discrete Analogues ◽  
Work Done ◽  
Potential Enstrophy

Abstract. The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids; and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos and others) and Discrete Exterior Calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp and others). Detailed results of the schemes applied to standard test cases are deferred to Part 2 of this series of papers.

scholarly journals Modified Predictor-Corrector WAF Method for the Shallow Water Equations with Source Terms

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Montri Maleewong
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Weighted Average ◽  
Test Cases ◽  
Half Time ◽  
Source Terms ◽  
Time Step ◽  
Predictor Corrector

A modified predictor-corrector scheme combining with the depth gradient method (DGM) and the weighted average flux (WAF) method has been presented to solve the one-dimensional shallow water equations with source terms. Approximate solutions in the predictor step are obtained by the DGM with piecewise-linear reconstructions in each cell volume. The source terms can then be calculated directly by these predicted values at the corresponding half-time step. In the corrector step, the TVD version of the WAF method is applied to calculate the numerical fluxes at the same half-time step for each cell face. The accuracy of numerical solutions is shown by applying the method to solve various test cases in both steady and unsteady problems with and without source terms. It shows that the numerical results are in good agreement with the existing analytical solutions as well as experimental data in some test cases.

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