Simulations Of Shallow Water Equations Research Articles

Impact of north-west Bay of Bengal bathymetry along the off-coast of Digha on the propagation of shallow water waves using finite difference method

Numerical simulation of shallow water equations has been carried out using the Crank-Nicolson scheme of Finite Difference Method (FDM) to study the interaction of waves with different forms of bathymetry especially that of the off-coast of Digha, a tourist spot situated in the north-west coast of Bay of Bengal. MATLAB code has been developed for the simulation. One-dimensional (1-D) linear shallow water model has been considered here. Three different bathymetry profile are chosen for the investigation: (i) flat bathymetry (with or without seamount/basin); (ii) piecewise linear approximation to the real bathymetry of the off-coast of Digha; (iii) real bathymetry of the off-coast of Digha. Real bathymetry data has been obtained from the gridded data set of GEBCO (General Bathymetric Chart of the Oceans) published in 2022. Validation of the present numerical scheme has been performed by considering test problems with results existing in the literature. Comparative study of the effect of the bathymetries, on the propagation of shallow water waves, has been performed and demonstrated with figures and tables. Results of wave-bathymetry interaction obtained here are believed to be new with respect to the present study area and will definitely give some insight into the phenomena in this region.

A well-balanced positivity-preserving multidimensional central scheme for shallow water equations

Recently, an efficient and multidimensional central scheme has been proposed by applying the limiting process diagonally (Yan et al. (2023) [25]). The aforementioned approach is applied to solve the two-dimensional shallow water equations with non-flat bottom topography in this study. The nonnegativity of the updated water depth is preserved under a reasonable CFL condition. The well-balancedness is achieved by simply applying central difference discretization to the source term using the surface gradient method. The conservation of water mass is obtained by changing the scheme into conservation form. All the key features of the present scheme have been rigorously proven. Several numerical experiments are provided to illustrate the robustness of the proposed scheme, including simulations of shallow water equations with Manning's bottom friction and Coriolis forces.

High-Order WENO Schemes with an Immersed Boundary Method for Shallow Water Equations on the Tsunami Mitigation with Configurations of Cylinder Array

In this paper, we investigated the flow around vertical circular cylinders with the numerical simulation of shallow water equations. Since tsunami tidal waves have large potential energy due to height difference, the man-made structures near coast are heavily damaged during the propagation of waves. For the damage mitigation of natural disaster, a careful configuration of such obstructive structures is worthy of attention, and it is of great practical importance to provide numerical simulation based on the ideal models of natural or artificial structures in a coastal area. The simplified numerical models are listed as the single large cylinder and four small cylinders of double inline, double staggered, and single four arrangement with fixing the same blockage ratio. Among various up-to-date schemes, the fifth-order WENO scheme is selected with a ghost-cell immersed boundary method, which enabled the easy detection of the arbitrary body shapes emerged in background Cartesian grids. In addition, we proposed the non-reflecting characteristic boundary condition based on the characteristic compatibility equations derived in shallow water equations to suppress the unexpected reflection waves at the open boundary. The analysis on a benchmark problem on solitary wave reflection led to an expected validation with the present numerical model. Finally, reflection and diffraction of a tsunami wave are analyzed against a single and four cylinders of the same blockage density, and it is believed to reduce 20.80% points for the best case of single four cylinders in the view of dissipation ratio defined by the kinematic power.

OpenMP Performance in Numerical Simulation of Dambreak Problem Using Shallow Water Equations

Numerical simulation of water surface waves is widely used to describe water flow and its impact for human life. For instance, numerical simulation of waves is elaborated to simulate Tsunami as an early warning system. Using numerical approach, the study of water flow will reduce cost and save time compared with the conventional approach (in laboratory). Shallow water equations (SWE) is one of the mathematical models which can be used to describe water flow. In numerical simulation of SWE, finite volume method is a robust method to approximate SWE. The result using numerical approach depends on the number of grids. High number of grids then smooth solution can be obtained. However, increasing number of grids leads to the increasing of computational cost. In this paper, parallel computing using OpenMP platform is given in order to reduce computational cost of numerical simulation. In parallel computing performances, Speedup andEfficiency of numerical simulation using 6400 grids points are obtained 4 times and 51% respectively.Moreover, by several numbers of cores from 2 to 8, CPU time of parallel computing is shown decreasing along the increasing number of computer cores.

Fast and accurate simulations of shallow water equations in large networks

We present a fast and accurate method for solving shallow water flows in large networks. The governing equations consist of the one-dimensional shallow water equations including friction forces. A class of first-order coupling conditions are implemented at the water intersections. Two-dimensional counterparts for simple water networks are also considered for comparison. As a numerical solver for shallow water equations we employed the discontinuous Galerkin method in one and two space dimensions. The method is simple, highly accurate and exhibits enhanced stability properties in the vicinity of sharp gradients which are often present in the solution of shallow water flow problems. The method belongs to a class of locally conservative finite element methods whose approximate solutions are discontinuous across inter-element boundaries. The performance of the proposed techniques is assessed on several applications for shallow water flows in networks. The aim of such a method compared to the conventional two-dimensional simulations is to solve shallow water flows in large networks efficiently and with an appropriate level of accuracy.

Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework

Lee, H. and Lee, N., 2018. Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 106–110. Coconut Creek (Florida), ISSN 0749-0208.The characteristic Galerkin finite element model for the discrete Boltzmann equation is developed for the depth-integrated shallow water equations. The lattice Boltzmann method (LBM) simulates microscopic collision and streaming of fluid particles and, then, evaluates the macroscopic velocity, pressure, etc. from the computed particle distribution. Due to the inherent geometric flexibility of the unstructured mesh in finite element method, the mesh clustering near large gradients and the numerical treatments of complex geometries has become possible. Some benchmark problems of practical importance (subcritical flow over hump, flow around cylinder, curved channel, etc.) are tested and the numerical results are in good agreement with previous experimental or analytic solutions.

Super- and sub-rotating equatorial jets in shallow water models of Jovian atmospheres: Newtonian cooling versus Rayleigh friction

Numerical simulations of the shallow water equations on rotating spheres produce mixtures of robust vortices and alternating zonal jets, as seen in the atmospheres of the gas giant planets. However, simulations that include Rayleigh friction invariably produce a sub-rotating (retrograde) equatorial jet for Jovian parameter regimes, whilst observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted over several decades. Super-rotating equatorial jets have recently been obtained in shallow water simulations that include a Newtonian relaxation of perturbations to the layer thickness to model radiative cooling to space, and in simulations of the thermal shallow water equations that include a similar relaxation term in their temperature equation. Simulations of global quasigeostrophic forms of these different models produce equatorial jets in the same directions as the parent models, suggesting that the mechanism responsible for setting the direction lies within quasigeostrophic theory. We provide such a mechanism by calculating the effective force acting on the thickness-weighted zonal mean flow due to the decay of an equatorially trapped Rossby wave. Decay due to Newtonian cooling creates an eastward zonal mean flow at the equator, consistent with the formation of a super-rotating equatorial jet, while decay due to Rayleigh friction leads to a westward zonal mean flow at the equator, consistent with the formation of a sub-rotating equatorial jet. In both cases the meridionally integrated zonal mean of the absolute zonal momentum is westward, consistent with the standard result that Rossby waves carry westward pseudomomentum, but this does not preclude the zonal mean flow being eastward on and close to the equator.

Well-balanced hybrid compact-WENO scheme for shallow water equations

We investigate the performance of the high order well-balanced hybrid compact-weighted essentially non-oscillatory (WENO) finite difference scheme (Hybrid) for simulations of shallow water equations with source terms due to a non-flat bottom topography. The Hybrid scheme employs the nonlinear fifth order characteristic-wise WENO-Z finite difference scheme to capture high gradients and discontinuities in an essentially non-oscillatory manner, and the linear spectral-like sixth order compact finite difference scheme to resolve the fine scale structures in the smooth regions of the solution efficiently and accurately. The high order multi-resolution analysis is employed to identify the smoothness of the solution at each grid point. In this study, classical one- and two-dimensional simulations, including a long time two-dimensional dam-breaking problem with a non-flat bottom topography, are conducted to demonstrate the performance of the hybrid scheme in terms of the exact conservation property (C-property), good resolution and essentially non-oscillatory shock capturing of the smooth and discontinuous solutions respectively, and up to 2–3 times speedup factor over the well-balanced WENO-Z scheme.

A modified Rusanov scheme for shallow water equations with topography and two phase flows

In this work, we introduce a finite volume method for numerical simulation of shallow water equations with source terms in one and two space dimensions, and one-pressure model of two-phase flows in one space dimension. The proposed method is composed of two steps. The first, called predictor step, depends on a local parameter allowing to control the numerical diffusion. A strategy based on limiters theory enables to control this parameter. The second step recovers the conservation equation. The scheme can thus be turned to order 1 in the regions where the flow has a strong variation, and order 2 in the regions where the flow is regular. The numerical scheme is applied to several test cases in one and two space dimensions. This scheme demonstrates its well-balanced property, and that it is an efficient and accurate approach for solving shallow water equations with and without source terms, and water faucet problem.

Dispersion Effects on Tsunami Propagation in South China Sea

As the Manila Trench is becoming the most tsunami-hazardous, it is necessary to ascertain the wave height and arrival time in the South China Sea region through numerical simulation of tsunamis generated from potential earthquake source along the Manila subduction zone. The Okada model is employed to generate tsunami. The surface elevation and depth-averaged horizontal velocity at first 5 min, coming from the simulation of shallow water equations, are then interpolated in the weakly dispersive model (FUNWAVE) to calculate tsunami propagation and far-field impact. The characteristics of tsunami wave height distribution in South China Sea are analyzed for the assessment of tsunami hazard near coasts around South China Sea due to the hypothetical earthquakes with magnitude of [Formula: see text] and the worst case scenario ([Formula: see text]). The maximum wave height distribution computed by the Boussinesq equations is compared with that by the shallow water equations to investigate the dispersion effects on propagation of tsunami in South China Sea. It is found that the dispersion effects of the tsunami waves propagating in South China Sea are not significant if the earthquake magnitude is large enough.

Thermal shallow water models of geostrophic turbulence in Jovian atmospheres

Conventional shallow water theory successfully reproduces many key features of the Jovian atmosphere: a mixture of coherent vortices and stable, large-scale, zonal jets whose amplitude decreases with distance from the equator. However, both freely decaying and forced-dissipative simulations of the shallow water equations in Jovian parameter regimes invariably yield retrograde equatorial jets, while Jupiter itself has a strong prograde equatorial jet. Simulations by Scott and Polvani [“Equatorial superrotation in shallow atmospheres,” Geophys. Res. Lett. 35, L24202 (2008)] have produced prograde equatorial jets through the addition of a model for radiative relaxation in the shallow water height equation. However, their model does not conserve mass or momentum in the active layer, and produces mid-latitude jets much weaker than the equatorial jet. We present the thermal shallow water equations as an alternative model for Jovian atmospheres. These equations permit horizontal variations in the thermodynamic properties of the fluid within the active layer. We incorporate a radiative relaxation term in the separate temperature equation, leaving the mass and momentum conservation equations untouched. Simulations of this model in the Jovian regime yield a strong prograde equatorial jet, and larger amplitude mid-latitude jets than the Scott and Polvani model. For both models, the slope of the non-zonal energy spectra is consistent with the classic Kolmogorov scaling, and the slope of the zonal energy spectra is consistent with the much steeper spectrum observed for Jupiter. We also perform simulations of the thermal shallow water equations for Neptunian parameter values, with a radiative relaxation time scale calculated for the same 25 mbar pressure level we used for Jupiter. These Neptunian simulations reproduce the broad, retrograde equatorial jet and prograde mid-latitude jets seen in observations. The much longer radiative time scale for the colder planet Neptune explains the transition from a prograde to a retrograde equatorial jet, while the broader jets are due to the deformation radius being a larger fraction of the planetary radius.

The effects of the spatial distribution of bottom topography and bottom drag on seiche-induced wave train formation

We consider the nonlinear, non-hydrostatic dynamics of seiches in small to medium-sized lakes. Using numerical simulations of shallow water equations modified to include weakly non-hydrostatic effects, we illustrate how spatially varying bottom drag and finite amplitude topography lead to the bending of wave trains that develop from the initial standing wave. For the case of variable topography, we discuss how the seiche and the wave trains that develop can resuspend material (e.g. nutrients) from the bottom of the lake and redistribute it in space. The numerical methods employed are spectrally accurate in space and second-order in time, yielding excellent accuracy and little numerical dissipation. We find that while the resuspension itself is largely due to the long standing waves at early times, the redistribution of nutrient distribution that is seen at later times is profoundly influenced by the development of the wave trains; a fundamentally non-hydrostatic effect.

Simulations of Shallow Water Equations by Finite Difference WENO Schemes with Multilevel Time Discretization

Simulations of Shallow Water Equations by Finite Difference WENO Schemes with Multilevel Time Discretization

Simulations of Shallow Water Equations with Finite Difference Lax-Wendroff Weighted Essentially Non-oscillatory Schemes

In this paper we study a Lax-Wendroff-type time discretization procedure for the finite difference weighted essentially non-oscillatory (WENO) schemes to solve one-dimensional and two-dimensional shallow water equations with source terms. In order to maintain genuinely high order accuracy and suit to problems with a rapidly varying bottom topography we use WENO reconstruction not only to the flux but also to the source terms of algebraical modified shallow water equations. Extensive simulations are performed, as a result, the WENO schemes with Lax-Wendroff-type time discretization can maintain nonoscillatory properties and more cost effective than that with Runge-Kutta time discretization.

Parallel hyperbolic PDE simulation on clusters: Cell versus GPU

Increasingly, high-performance computing is looking towards data-parallel computational devices to enhance computational performance. Two technologies that have received significant attention are IBM's Cell Processor and NVIDIA's CUDA programming model for graphics processing unit (GPU) computing. In this paper we investigate the acceleration of parallel hyperbolic partial differential equation simulation on structured grids with explicit time integration on clusters with Cell and GPU backends. The message passing interface (MPI) is used for communication between nodes at the coarsest level of parallelism. Optimizations of the simulation code at the several finer levels of parallelism that the data-parallel devices provide are described in terms of data layout, data flow and data-parallel instructions. Optimized Cell and GPU performance are compared with reference code performance on a single x86 central processing unit (CPU) core in single and double precision. We further compare the CPU, Cell and GPU platforms on a chip-to-chip basis, and compare performance on single cluster nodes with two CPUs, two Cell processors or two GPUs in a shared memory configuration (without MPI). We finally compare performance on clusters with 32 CPUs, 32 Cell processors, and 32 GPUs using MPI. Our GPU cluster results use NVIDIA Tesla GPUs with GT200 architecture, but some preliminary results on recently introduced NVIDIA GPUs with the next-generation Fermi architecture are also included. This paper provides computational scientists and engineers who are considering porting their codes to accelerator environments with insight into how structured grid based explicit algorithms can be optimized for clusters with Cell and GPU accelerators. It also provides insight into the speed-up that may be gained on current and future accelerator architectures for this class of applications. Program summary Program title: SWsolver Catalogue identifier: AEGY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGY_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL v3 No. of lines in distributed program, including test data, etc.: 59 168 No. of bytes in distributed program, including test data, etc.: 453 409 Distribution format: tar.gz Programming language: C, CUDA Computer: Parallel Computing Clusters. Individual compute nodes may consist of x86 CPU, Cell processor, or x86 CPU with attached NVIDIA GPU accelerator. Operating system: Linux Has the code been vectorised or parallelized?: Yes. Tested on 1-128 x86 CPU cores, 1-32 Cell Processors, and 1-32 NVIDIA GPUs. RAM: Tested on Problems requiring up to 4 GB per compute node. Classification: 12 External routines: MPI, CUDA, IBM Cell SDK Nature of problem: MPI-parallel simulation of Shallow Water equations using high-resolution 2D hyperbolic equation solver on regular Cartesian grids for x86 CPU, Cell Processor, and NVIDIA GPU using CUDA. Solution method: SWsolver provides 3 implementations of a high-resolution 2D Shallow Water equation solver on regular Cartesian grids, for CPU, Cell Processor, and NVIDIA GPU. Each implementation uses MPI to divide work across a parallel computing cluster. Additional comments: Sub-program numdiff is used for the test run.