A Weighted-Least-Squares Meshless Model for Non-Hydrostatic Shallow Water Waves

In this paper, an explicit time marching procedure for solving the non-hydrostatic shallow water equation (SWE) problems is developed. The procedure includes a process of prediction and several iterations of correction. In these processes, it is essential to accurately calculate the spatial derives of the physical quantities such as the temporal water depth, the average velocities in the horizontal and vertical directions, and the dynamic pressure at the bottom. The weighted-least-squares (WLS) meshless method is employed to calculate these spatial derivatives. Though the non-hydrostatic shallow water equations are two dimensional, on the focus of presenting this new time marching approach, we just use one dimensional benchmark problems to validate and demonstrate the stability and accuracy of the present model. Good agreements are found in the comparing the present numerical results with analytic solutions, experiment data, or other numerical results.

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A Coupled-Mode Technique for the Run-Up of Non-Breaking Dispersive Waves on Plane Beaches

Volume 2: Ocean Engineering and Polar and Arctic Sciences and Technology ◽

10.1115/omae2006-92162 ◽

2006 ◽

Author(s):

K. A. Belibassakis ◽

G. A. Athanassoulis

Keyword(s):

Shallow Water ◽

Water Waves ◽

Wave Equations ◽

Neumann Boundary ◽

Breaking Waves ◽

Numerical Results ◽

Coupled Mode Theory ◽

Mode Theory ◽

Coupled Mode ◽

Run Up

A coupled-mode model is developed and applied to the transformation and run-up of dispersive water waves on plane beaches. The present work is based on the consistent coupled-mode theory for the propagation of water waves in variable bathymetry regions, developed by Athanassoulis & Belibassakis (1999) and extended to 3D by Belibassakis et al (2001), which is suitably modified to apply to a uniform plane beach. The key feature of the coupled-mode theory is a complete modal-type expansion of the wave potential, containing both propagating and evanescent modes, being able to consistently satisfy the Neumann boundary condition on the sloping bottom. Thus, the present approach extends previous works based on the modified mild-slope equation in conjunction with analytical solution of the linearised shallow water equations, see, e.g., Massel & Pelinovsky (2001). Numerical results concerning non-breaking waves on plane beaches are presented and compared with exact analytical solutions; see, e.g., Wehausen & Laitone (1960, Sec. 18). Also, numerical results are presented concerning the run-up of non-breaking solitary waves on plane beaches and compared with the ones obtained by the solution of the shallow-water wave equations, Synolakis (1987), Li & Raichlen (2002), and experimental data, Synolakis (1987).

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Validation of Depth-Averaged Flow Model Using Flat-Bottomed Benchmark Problems

The Scientific World JOURNAL ◽

10.1155/2014/197539 ◽

2014 ◽

Vol 2014 ◽

pp. 1-18 ◽

Cited By ~ 4

Author(s):

Il Won Seo ◽

Young Do Kim ◽

Chang Geun Song

Keyword(s):

Shallow Water ◽

Mass Conservation ◽

Analytic Solutions ◽

Benchmark Problems ◽

Roughness Coefficient ◽

Spanwise Direction ◽

Clear Trend ◽

Recirculating Flow ◽

Fully Implicit ◽

Algorithmic Approaches

In this study, a shallow water flow code was developed and tested against four benchmark problems of practical relevance. The results demonstrated that as the eddy viscosity increased, the velocity slope along the spanwise direction decreased, and the larger roughness coefficient induced a higher flow depth over the channel width. The mass conservation rate was determined to be 99.2%. This value was measured by the variation of the total volume of the fluid after a cylinder break. As the Re increased to 10,000 in the internal recirculating flow problem, the intensity of the primary vortex had a clear trend toward the theoretically infinite Re value of −1.886. The computed values of the supercritical flow evolved by the oblique hydraulic jump agreed well with the analytic solutions within an error bound of 0.2%. The present model adopts the nonconservative form of shallow water equations. These equations are weighted by the SU/PG scheme and integrated by a fully implicit method, which can reproduce physical problems with various properties. The model provides excellent results under various flow conditions, and the solutions of benchmark tests can present criteria for the evaluation of various algorithmic approaches.

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Nonlinear Shallow-Water Flow on Deck

Journal of Ship Research ◽

10.5957/jsr.1996.40.4.303 ◽

1996 ◽

Vol 40 (04) ◽

pp. 303-315

Author(s):

Zhenjia Huang ◽

Chi-Chao Hsiung

Keyword(s):

Shallow Water ◽

Water Flow ◽

Shallow Water Equation ◽

Splitting Method ◽

Numerical Results ◽

Governing Equations ◽

Shallow Water Flow ◽

Step Method ◽

Fractional Step Method ◽

Flux Difference

Euler's equations have been used for nonlinear shallow-water flow on deck. The equations are simplified under the shallow-water assumption to obtain the governing equations. The Flux-Difference Splitting method is devised to solve this shallow-water flow problem. The flux-difference in the governing equations is split based on characteristic directions. The Superbee flux limiter is employed in the algorithm to make the finite-difference scheme of second order with high resolution. For two-dimensional decks, numerical results are presented to reveal the characteristics of shallowwater flow on deck. For three-dimensional decks, the Flux-Difference Splitting method together with the Fractional Step method are used, so that solutions of the shallow-water equation can be obtained by solving two sets of one-dimensional differential equations. Numerical results are presented to show the wave patterns for different modes of motion excitation. Velocity vectors in the flow field are also given to help understand the flow properties.

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Reflection of a shallow-water soliton. Part 1. Edge layer for shallow-water waves

Journal of Fluid Mechanics ◽

10.1017/s0022112084001919 ◽

1984 ◽

Vol 146 ◽

pp. 369-382 ◽

Cited By ~ 16

Author(s):

N. Sugimoto ◽

T. Kakutani

Keyword(s):

Boundary Condition ◽

Shallow Water ◽

Water Waves ◽

Shallow Water Equation ◽

Expansion Method ◽

Matching Condition ◽

Finite Amplitude ◽

Matched Asymptotic Expansion ◽

Dispersive Waves ◽

Edge Layer

To investigate reflection of a shallow-water soliton at a sloping beach, the edge-layer theory is developed to obtain a ‘reduced’ boundary condition relevant to the simplified shallow-water equation describing the weakly dispersive waves of small but finite amplitude. An edge layer is introduced to take account of the essentially two-dimensional motion that appears in the narrow region adjacent to the beach. By using the matched-asymptotic-expansion method, the edge-layer theory is formulated to cope with the shallow-water theory in the offshore region and the boundary condition at the beach. The ‘reduced’ boundary condition is derived as a result of the matching condition between the two regions. An explicit edge-layer solution is obtained on assuming a plane beach.

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

E-Jurnal Matematika ◽

10.24843/mtk.2018.v07.i02.p190 ◽

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.

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Shallow-Water-Equation Model for Simulation of Earthquake-Induced Water Waves

Mathematical Problems in Engineering ◽

10.1155/2017/3252498 ◽

2017 ◽

Vol 2017 ◽

pp. 1-11

Author(s):

Hongzhou Ai ◽

Lingkan Yao ◽

Haixin Zhao ◽

Yiliang Zhou

Keyword(s):

Finite Difference Method ◽

Shallow Water ◽

Water Waves ◽

Seismic Waves ◽

Empirical Equation ◽

Shallow Water Equation ◽

Equation Model ◽

Proposed Model ◽

Maximum Water ◽

Water Elevation

A shallow-water equation (SWE) is used to simulate earthquake-induced water waves in this study. A finite-difference method is used to calculate the SWE. The model is verified against the models of Sato and of Demirel and Aydin with three kinds of seismic waves, and the numerical results of earthquake-induced water waves calculated using the proposed model are reasonable. It is also demonstrated that the proposed model is reliable. Finally, an empirical equation for the maximum water elevation of earthquake-induced water waves is developed based on the results obtained using the model, which is an improvement on former models.

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Wave-current interactions representation by coupling spectral wave and coastal hydrodynamics models

10.5194/egusphere-egu21-286 ◽

2021 ◽

Author(s):

Anastasia Fragkou ◽

Christopher Old ◽

Athanasios Angeloudis

Keyword(s):

Shallow Water ◽

Analytical Solutions ◽

Water Waves ◽

Shallow Water Equation ◽

Computational Cost ◽

Coupled Model ◽

Wave Model ◽

Geophysical Research ◽

Spectral Wave Model ◽

Coastal Hydrodynamics

A parallelized unstructured coupled model is developed to investigate wave-current interactions in coastal waters at regional scales. This model links the spectral wave model Simulating Waves Nearshore (SWAN; Booij et al., 1999) with the coastal hydrodynamics shallow-water equation model Thetis (K&#228;rn&#228; et al., 2018). SWAN is based on the action density equations encompassing the various source-terms accounting for deep- and shallow-water phenomena. Thetis solves the non-conservative form of the depth-averaged shallow water equations implemented within Firedrake, an abstract framework for the solution of Finite Element Method (FEM) problems. In resolving wave-current interactions in the proposed model, Thetis predicts water elevation and current velocities which are communicated in SWAN, while the latter provides radiation stresses information for the former. The numerical domain is prescribed by an unstructured mesh allowing higher resolution to areas of interest, while maintaining a reasonable computational cost. As the models share the same mesh, interpolation errors and certain computational overheads can be contained, whereas the choice to employ a sub-mesh for SWAN model is being considered to reduce the overall cost.The model is initially validated and its performance assessed by a slowly varying-bathymetry. Predictions are compared against the analytical solutions for the wave setup and significant wave height (Longuet-Higgins and Stewart, 1964). Comparisons also extend to results from a coupled 3-D hydrodynamics model with a spectral wave model (Roland et al., 2012). The results of the proposed coupled model exhibit good correlations with the analytical solutions showcasing the same level of e&#64259;ciency as the 3-D coupled model.&#160;References[1] Booij N, Ris RC, Holthuijsen LH. A third-generation wave model for coastal regions: 1. Model&#160;description and validation. Journal of geophysical research: Oceans 1999;104(C4):7649&#8211;7666.[2] K&#228;rn&#228; T, Kramer SC, Mitchell L, Ham DA, Piggott MD, Baptista AM. Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations. Geoscientific Model Development 2018;11(11):4359&#8211;4382.[3] Longuet-Higgins MS, Stewart R. Radiation stresses in water waves; a physical discussion, with applications. In: Deep sea research and oceanographic abstracts, vol. 11 Elsevier; 1964. p. 529&#8211;562.[4] Roland A, Zhang YJ, Wang HV, Meng Y, Teng YC, Maderich V, et al. A fully coupled 3D wave-current interaction model on unstructured grids. Journal of Geophysical Research: Oceans 2012;117(C11).

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Time and Frequency Domain Analyses of Shallow Water Waves on a Slope

Coastal Engineering 1990 ◽

10.1061/9780872627765.024 ◽

1991 ◽

Author(s):

D H Swart ◽

J B Crowley

Keyword(s):

Shallow Water ◽

Frequency Domain ◽

Water Waves ◽

Shallow Water Waves

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Evaluation for estimating of the PDF and the CDF of Generalized Inverted Exponential Distribution with Application in Industry

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.39 ◽

2020 ◽

pp. 507-522

Author(s):

Parisa Torkaman

Keyword(s):

Least Squares ◽

Exponential Distribution ◽

Mean Squared Error ◽

Weighted Least Squares ◽

Real Data ◽

Minimum Variance ◽

Cumulative Distribution ◽

Estimation Methods ◽

Data Set ◽

Better Than

The generalized inverted exponential distribution is introduced as a lifetime model with good statistical properties. This paper, the estimation of the probability density function and the cumulative distribution function of with five different estimation methods: uniformly minimum variance unbiased(UMVU), maximum likelihood(ML), least squares(LS), weighted least squares (WLS) and percentile(PC) estimators are considered. The performance of these estimation procedures, based on the mean squared error (MSE) by numerical simulations are compared. Simulation studies express that the UMVU estimator performs better than others and when the sample size is large enough the ML and UMVU estimators are almost equivalent and efficient than LS, WLS and PC. Finally, the result using a real data set are analyzed.

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