scholarly journals Fast Spherical Harmonic Transform Routine FLTSS Applied to the Shallow Water Test Set

2005 ◽  
Vol 133 (3) ◽  
pp. 634-648 ◽  
Author(s):  
Reiji Suda
Keyword(s):  
Computational Complexity ◽  
Shallow Water ◽  
Spherical Harmonic ◽  
Numerical Solutions ◽  
Shallow Water Equation ◽  
Approximate Algorithm ◽  
Water Model ◽  
Legendre Transform ◽  
Test Set ◽  
Spectral Transform

Abstract The fast spherical harmonic transform algorithm proposed by Suda and Takami is evaluated in the solutions of the shallow water equation test set defined by Williamson et al. through replacing the Legendre transforms of the NCAR spectral transform shallow water model (STSWM) with routines of the fast Legendre transform with stable sampling (FLTSS), which is the first implementation of the Suda–Takami algorithm. The Suda–Takami algorithm is an approximate algorithm with the computational complexity O(T 2 log T log ɛ−1), with T being the maximum wavenumber and ɛ the accuracy parameter of the FLTSS. The influence of the approximation errors of the FLTSS upon the numerical solutions is investigated. For all test cases of the Williamson et al. test set, the FLTSS stably solved the equations with the results that can be explained well with the accuracy ɛ. The stability in longer time integrations is also assessed, where test case 7 of an analyzed atmospheric initial condition was stably integrated for 1 yr. The FLTSS was faster than the STSWM at T170 and had higher resolutions on an Intel Mobile Pentium 4, where the lower space complexity (memory requirements) of the FLTSS was advantageous in addition to the lower computational complexity.

scholarly journals Congested shallow water model: roof modeling in free surface flow

2018 ◽  
Vol 52 (5) ◽  
pp. 1679-1707 ◽  
Author(s):  
Edwige Godlewski ◽  
Martin Parisot ◽  
Jacques Sainte-Marie ◽  
Fabien Wahl
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hyperbolic Equations ◽  
Mechanical Energy ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

Spectral Transform Solutions to the Shallow Water Test Set

1995 ◽  
Vol 119 (1) ◽  
pp. 164-187 ◽  
Author(s):  
Ruediger Jakob-Chien ◽  
James J. Hack ◽  
David L. Williamson
Keyword(s):  
Shallow Water ◽  
Test Set ◽  
Spectral Transform ◽  
Water Test

Axisymmetric gravity currents in a rotating system: experimental and numerical investigations

2001 ◽  
Vol 447 ◽  
pp. 1-29 ◽  
Author(s):  
MARK A. HALLWORTH ◽  
HERBERT E. HUPPERT ◽  
MARIUS UNGARISH
Keyword(s):  
Flow Field ◽  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Salt Water ◽  
Gravity Currents ◽  
The Body ◽  
Water Model ◽  
Maximum Radius ◽  
Theoretical Predictions

The propagation at high Reynolds number of a heavy, axisymmetric gravity current of given initial volume over a horizontal boundary is considered in both rotating and non-rotating situations. The investigation combines experiments with theoretical predictions by both shallow-water approximations and numerical solutions of the full axisymmetric equations. Attention is focused on cases when the initial ratio of Coriolis to inertia forces is small. The experiments were performed by quickly releasing a known cylindrical volume of dense salt water of 2 m diameter at the centre of a circular tank of diameter 13 m containing fresh ambient water of typical depth 80 cm. The propagation of the current was recorded for different initial values of the salt concentration, the volume of released fluid, the ratio of the initial height of the current to the ambient depth, and the rate of rotation. A major feature of the rotating currents was the attainment of a maximum radius of propagation. Thereafter a contraction–relaxation motion of the body of fluid and a regular series of outwardly propagating pulses was observed. The frequency of these pulses is slightly higher than inertial, and the amplitude is of the order of magnitude of half the maximum radius. Theoretical predictions of the corresponding gravity currents were also obtained by (i) previously developed shallow-water approximations (Ungarish & Huppert 1998) and (ii) a specially developed finite-difference code based on the full axisymmetric Navier–Stokes equations. The ‘numerical experiments’ provided by this code are needed to capture details of the flow field (such as the non-smooth shape of the interface, the vertical dependence of the velocity field) which are not reproduced by the shallow-water model and are very difficult for, or outside the range of, accurate experimental measurement. The comparisons and discussion provide insight into the flow field and indicate the advantages and limitations of the verified simulation tools.

scholarly journals A Discontinuous Galerkin Global Shallow Water Model

2005 ◽  
Vol 133 (4) ◽  
pp. 876-888 ◽  
Author(s):  
Ramachandran D. Nair ◽  
Stephen J. Thomas ◽  
Richard D. Loft
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Time Integration ◽  
Curvilinear Coordinates ◽  
Basis Set ◽  
Water Model ◽  
Test Case ◽  
Shallow Water Model ◽  
Cubed Sphere

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax–Friedrichs scheme. A third-order total variation diminishing Runge–Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

Efficient modeling of shallow water equations using method of lines and artificial viscosity

2019 ◽  
Vol 34 (04) ◽  
pp. 2050051
Author(s):  
Mohamed M. Mousa ◽  
Wen-Xiu Ma
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Method Of Lines ◽  
Artificial Viscosity ◽  
Water Model ◽  
Numerical Schemes ◽  
Von Neumann ◽  
The Method Of Lines ◽  
Shallow Water Models ◽  
Von Neumann Stability

In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results.

scholarly journals KONVERGENSI NUMERIK FLUKS RUSANOV DAN HLLE PADA METODE BEDA VOLUM UNTUK MENGHAMPIRI PERSAMAAN AIR DANGKAL

2018 ◽  
Vol 7 (2) ◽  
pp. 94
Author(s):  
EKO MEIDIANTO N. R. ◽  
P. H. GUNAWAN ◽  
A. ATIQI ROHMAWATI
Keyword(s):  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Water Waves ◽  
Numerical Solutions ◽  
Shallow Water Equation ◽  
Average Error ◽  
One Dimensional ◽  
Dimensional Simulation ◽  
Volume Method

This one-dimensional simulation is performed to find the convergence of different fluxes on the water wave using shallow water equation. There are two cases where the topography is flat and not flat. The water level and grid of each simulation are made differently for each case, so that the water waves that occur can be analyzed. Many methods can be used to approximate the shallow water equation, one of the most used is the finite volume method. The finite volume method offers several numerical solutions for approximate shallow water equation, including Rusanov and HLLE. The derivation result of the numerical solution is used to approximate the shallow water equation. Differences in numerical and topographic solutions produce different waves. On flat topography, the rusanov flux has an average error of 0.06403 and HLLE flux with an average error of 0.06163. While the topography is not flat, the rusanov flux has a 1.63250 error and the HLLE flux has an error of 1.56960.

scholarly journals INTER-COUPLED TSUNAMI MODELLING THROUGH AN ABSORBING-GENERATING BOUNDARY

2020 ◽  
pp. 38
Author(s):  
Sangyoung Son ◽  
Patrick Lynett
Keyword(s):  
Shallow Water ◽  
Shallow Water Equation ◽  
Boussinesq Equations ◽  
Equation Model ◽  
Water Model ◽  
Navier Stokes ◽  
Characteristic Form ◽  
Computationally Efficient ◽  
Boussinesq Model ◽  
Tsunami Modelling

For many practical and theoretical purposes, various types of tsunami wave models have been developed and utilized so far. Some distinction among them can be drawn based on governing equations used by the model. Shallow water equations and Boussinesq equations are probably most typical ones among others since those are computationally efficient and relatively accurate compared to 3D Navier-Stokes models. From this idea, some coupling effort between Boussinesq model and shallow water equation model have been made (e.g., Son et al. (2011)). In the present study, we couple two different types of tsunami models, i.e., nondispersive shallow water model of characteristic form(MOST ver.4) and dispersive Boussinesq model of non-characteristic form(Son and Lynett (2014)) in an attempt to improve modelling accuracy and efficiency.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/cTXybDEnfsQ

scholarly journals SHALLOW WATER EQUATION SOLUTION IN 2D USING FINITE DIFFERENCE METHOD WITH EXPLICIT SCHEME

2017 ◽  
Vol 17 (2) ◽  
pp. 105
Author(s):  
Nuraini Nuraini ◽  
Syamsul Rizal ◽  
Marwan Marwan
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Shallow Water Equation ◽  
Mass Conservation ◽  
Conservation Equation ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
Three Dimensional Model

Abstract. Modeling the dynamics of seawater typically uses a shallow water model. The shallow water model is derived from the mass conservation equation and the momentum set into shallow water equations. A two-dimensional shallow water equation alongside the model that is integrated with depth is described in numerical form. This equation can be solved by finite different methods either explicitly or implicitly. In this modeling, the two dimensional shallow water equations are described in discrete form using explicit schemes.Keyword: shallow water equation, finite difference and schema explisit.REFERENSI 1. Bunya, S., Westerink, J. J. dan Yoshimura. 2005. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids. 47: 1451-1468.2. Kampf Jochen. 2009. Ocean Modelling For Beginners. Springer Heidelberg Dordrecht. London New York.3. Rezolla, L 2011. Numerical Methods for the Solution of Partial Diferential Equations. Trieste. International Schoolfor Advanced Studies.4. Natakussumah, K. D., Kusuma, S. B. M., Darmawan, H., Adityawan, B. M. Dan  Farid, M. 2007. Pemodelan Hubungan Hujan dan Aliran Permukaan pada Suatu DAS  dengan Metode Beda Hingga. ITB Sain dan Tek. 39: 97-123.5. Casulli, V. dan Walters, A. R. 2000. An unstructured grid, three-dimensional model based on the shallow water equations. Int. J. Numer. Meth. Fluids. 32: 331-348.6. Triatmodjo, B. 2002. Metode Numerik  Beta Offset. Yogyakarta.

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