On the representation of gravity waves in numerical models of the shallow-water equations

2000 ◽  
Vol 126 (563) ◽  
pp. 669-688 ◽  
Author(s):  
A. R. Mohebalhojeh ◽  
D. G. Dritschel
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Numerical Models

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.

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.

scholarly journals Inertia–gravity waves generated by near balanced flow in 2-layer shallow water turbulence on the β-plane

2013 ◽  
Vol 20 (1) ◽  
pp. 25-34 ◽  
Author(s):  
A. Wirth
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Continuous Emission ◽  
Coriolis Parameter ◽  
Balanced Flow ◽  
Grid Points ◽  
Fine Resolution ◽  
Enstrophy Cascade ◽  
Inertia Gravity Waves

Abstract. Using a fine resolution numerical model (40002 × 2 grid-points) of the two-layer shallow-water equations of the mid-latitude β-plane dynamics, it is shown that there is no sudden breakdown of balance in the turbulent enstrophy cascade but a faint and continuous emission of inertia–gravity waves. The wave energy accumulates in the equator-ward region of the domain due to the Coriolis parameter depending on the latitude and dispersion relation of inertia–gravity waves.

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 Multiple Solutions for the Riemann Problem in the Porous Shallow Water Equations

10.29007/31n4 ◽  
2018 ◽  
Author(s):  
Luca Cozzolino ◽  
Raffaele Castaldo ◽  
Luigi Cimorelli ◽  
Renata Della Morte ◽  
Veronica Pepe ◽  
...  
Keyword(s):  
Shallow Water ◽  
Riemann Problem ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Numerical Models ◽  
Control Volume ◽  
Fluid Phase ◽  
Problem Solution ◽  
The One ◽  
Solid Obstacles

The Porous Shallow water Equations are widely used in the context of urban flooding simulation. In these equations, the solid obstacles are implicitly taken into account by averaging the classic Shallow water Equations on a control volume containing the fluid phase and the obstacles. Numerical models for the approximate solution of these equations are usually based on the approximate calculation of the Riemann fluxes at the interface between cells. In the present paper, it is presented the exact solution of the one-dimensional Riemann problem over the dry bed, and it is shown that the solution always exists, but there are initial conditions for which it is not unique. The non-uniqueness of the Riemann problem solution opens interesting questions about which is the physically congruent wave configuration in the case of solution multiplicity.

scholarly journals SEMI-IMPLICIT NUMERICAL SCHEMA IN SHALLOW WATER EQUATION

2017 ◽  
Vol 17 (2) ◽  
pp. 102
Author(s):  
Safwandi Safwandi ◽  
Syamsul Rizal ◽  
Tarmizi Tarmizi
Keyword(s):  
New York ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Shallow Water Equation ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Spectral Difference ◽  
Spectral Difference Method

Abstract. A two-dimensional shallow water equation integrated on depth water based on finite differential methods. Numerical solutions with different methods consist of explicit, implicit and semi-implicit schemes. Different methods of shallow water equations expressed in numerical schemes. For bottom-friction is described in semi-implicitly. This scheme will be more flexible for initial values and boundary conditions when compared to the explicit schemes.  Keywords: 2D numerical models, shallow water equations, explicit and semi-implicit schema.Reference Hassan, H. S., Ramadan, K. T., Hanna, S. N. 2010. Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method, Scie.Res,App.Math. (1):104-117. Omer, S, Kursat, K. 2011. High-Order Accurate Spectral Difference Method For Shallow Water Equations. IJRRAS6. Vol. 6. No. 1. Kampf, J. 2009. Ocean Modelling for Beginners. Springer Heidelberg Dordrecht. London, New York. Wang, Z. L., Geng, Y. F. 2013. Two-Dimensional Shallow Water Equations with Porosity and Their Numerical scheme on Unstructured Grids. J. Water Science and Engineering. Vol. 6, No. 1, 91-105. Saiduzzaman, Sobuj. 2013. Comparison of Numerical Schemes for Shallow Water Equation. Global J. of Sci. Fron. Res. Math. and Dec. Sci. Vol. 13 (4). Sari, C. I., Surbakti, H., Fauziyah., Pola Sebaran Salinatas dengan Model Numerik Dua Dimensi di Muara Sungai Musi. Maspari J. Vol. 5 (2): 104-110. Bunya, B., Westerink, J. J. dan Shinobu, Y. 2004. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids 2005 (47): 1451–1468. 

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