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 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.

A General Method for Conserving Energy and Potential Enstrophy in Shallow-Water Models

2007 ◽  
Vol 64 (2) ◽  
pp. 515-531 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Models ◽  
Finite Difference Approximations ◽  
Nambu Bracket ◽  
Shallow Water Models ◽  
Explicit Finite Difference ◽  
Mass Circulation ◽  
General Method ◽  
Potential Enstrophy

Abstract The shallow-water equations may be posed in the form df /dt = {F, H, Z}, where H is the energy, Z is the potential enstrophy, and the Nambu bracket {F, H, Z} is completely antisymmetric in its three arguments. This makes it very easy to construct numerical models that conserve analogs of the energy and potential enstrophy; one need only discretize the Nambu bracket in such a way that the antisymmetry property is maintained. Using this strategy, this paper derives explicit finite-difference approximations to the shallow-water equations that conserve mass, circulation, energy, and potential enstrophy on a regular square grid and on an unstructured triangular mesh. The latter includes the regular hexagonal grid as a special case.

Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid

2020 ◽  
Vol 35 (6) ◽  
pp. 355-366
Author(s):  
Vladimir V. Shashkin ◽  
Gordey S. Goyman
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Time Integration ◽  
Standard Test ◽  
Integration Scheme ◽  
Test Problems ◽  
Cubed Sphere ◽  
Lagrangian Scheme ◽  
Inertia Gravity Waves

AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

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.

Application of the TELEMAC-2D Model in the Fluvial Hydrodinamics Simulation and Reproduction of Flood Patterns

2019 ◽  
Vol 396 ◽  
pp. 187-196
Author(s):  
Aldair Forster ◽  
Juliana Costi ◽  
Wiliam Correa Marques ◽  
André Gustavo Wormsbecher ◽  
Antonio Raylton Rodrigues Bendo
Keyword(s):  
Finite Element ◽  
High Resolution ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Models ◽  
Finite Element Mesh ◽  
Element Mesh ◽  
The City ◽  
2D Model ◽  
National Water

. The increased occurrence of floods in the city of Rio do Sul (SC), even with the creation of dams to contain floods, show that non-structural measures can be good alternatives to reduce losses in the region. Numerical flood modeling has been widely used to anticipate risks and assist in decisionmaking. One of the numerical models that is being used to simulate floods is TELEMAC-2D, which is able to simulate the hydrodynamics of open channels by solving the shallow water equations in a domain discretized by an unstructured finite element mesh. We used the TELEMAC-2D model tosimulate the dynamics of the rivers of the region of Rio do Sul throughout the year of 2013, period during which a flood with large proportions occurred in September. Fluviometric data avaliable from the National Water Agency and high resolution (1 m) topographic data provided by government agen-cies of Santa Catarina were used in the simulation. The results show that the model performed well in simulating the maximum flood extension occurred in September, however, the simulations were underestimated for most of the time, indicating that calibrations in the model can still be performed.

A standard test set for numerical approximations to the shallow water equations in spherical geometry

1992 ◽  
Vol 101 (1) ◽  
pp. 227-228 ◽  
Author(s):  
David L Williamson ◽  
John B Drake ◽  
James J Hack ◽  
Rüdiger Jakob ◽  
Paul N Swarztrauber
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Spherical Geometry ◽  
Standard Test ◽  
Numerical Approximations ◽  
Test Set
Sign in / Sign up

Export Citation Format

Share Document