1. Introduction

The usage of the shallow water models (SWMs) allows to solve a wide range of engineering tasks related to the dynamics of surface waters (a seasonal flooding [1], a drain and rain flows [2]), the emergence and expansion of the marine nonlinear waves [3] (problems of tsunami impact on a shore, nonlinear waves formation due to earthquakes, and meteorological waves generation by an open ocean resonance [4]), and the inundation in a coastal area by storm surge [5]. The SWM modifications are effective for studying various geophysical problems such as the dynamics of the pyroclastic flows [6] and the riverbed processes including the sediment dynamics and the diffusion of pollutant particles in reservoirs. The multilayer models utilization significantly expands the opportunities of the shallow water approach [7]. The tasks associated with flooding research of river valleys or interfluves [1] are stood out among the hydrological problems. In the framework of the SWM, the important results have

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

also been obtained for the atmospheric phenomena, meteorological forecasts, and global climate models [8, 9].

water equations, are considered. The SWE or the Saint-Venant equations are fairly simple model describing the free surface dynamics of incompressible fluid (see the details in the review of [23]). The model assumes that vertical equilibrium in a medium exists at every moment of time. The conservation laws of mass and momentum in the integral form for a thin

Hv dS ¼ �g

where Q is the sources function, ∇ = ex∂/∂x + ey∂/∂y is the nabla operator, S(t) is the crosssectional area of the "liquid particle", r = xe<sup>x</sup> + ye<sup>y</sup> is the radius vector, v = ue<sup>x</sup> + ve<sup>y</sup> is the velocity vector, η(r, t) = H(r, t) + b(r) is the free surface elevation, b(r) is the bottom profile, f(r, t) = f<sup>b</sup> + ffr + fCor + f<sup>w</sup> + f<sup>s</sup> is the sum of the external forces, accounting for the bottom friction fb, the internal friction (viscosity) ffr, the Coriolis force fCor, the effect of atmospheric wind fw, and the force f<sup>s</sup> determined by the liquid momentum due to the action of the sources Q. The

(rain, melting snow, flows through the hydro-constructions, groundwater, etc.) and sinks

The digital elevation model (DEM) provides the quality of numerical simulations of real hydrological objects to a significant extent. The DEM is determined by the height matrix bij = b(xi, yj) on numerical grid {xi, yj} (i = 1, …, Nx, j = 1, …, Ny). The DEM elaboration utilizes diverse geoinformation methods for the processing of spatial data obtained from various sources. A matrix of heights has been built in several stages. At the beginning stage, the remote sensing data are accounted by the function b(xi, yj). The river sailing directions and the actual water depth measurements allow to construct the DEM for large river beds (for example, for the Volga River and the Akhtuba River). To improve our DEM, the data on various small topography objects such as small waterways, roads, small dams, etc., should be included into the consideration. The numerical simulation results of the shallow water dynamics reveal flooding areas which may be compared with real observational data (both from the remote methods and our own GPS measurements). Such approach qualitatively improves the DEM as

In current paragraph, the CSPH-TVD numerical method is thoroughly examined. The method combines classical TVD and SPH methods at various stages of numerical integration of hyperbolic partial differential equations and uses the particular benefits of both. The advantages of graphical processing units (GPUs) with CUDA acceleration for hydrological simulations are

The computational domain has been covered by a fixed uniform grid with a spatial step h, while mobile "liquid particles" (hereinafter particles) are placed in the centers of cells. The time layers tn have a nonuniform step τ<sup>n</sup> = tn � tn � 1, where n denotes the index of the time layer. The vector space index i = (i, j) characterizes the radius vectors of the fixed centers of the cells

ðð

H∇η dS þ

A Numerical Simulation of the Shallow Water Flow on a Complex Topography

ðð

http://dx.doi.org/10.5772/intechopen.71026

Hf dS, (1)

239

are the liquid sources

S tð Þ

S tð Þ

dt ðð S tð Þ

layer of moving substance with additional sources and forces are:

surface density of sources is σ(r, t) = σ(+) + σ(-) = dQ/dS, where σ(+) and σ(�)

<sup>H</sup>ð Þ <sup>r</sup>; <sup>t</sup> dS <sup>¼</sup> <sup>Q</sup>ð Þ <sup>r</sup>; <sup>t</sup> , <sup>d</sup>

d dt ðð S tð Þ

(infiltration, evaporation, etc.), respectively.

a result of the iterative topography refinement.

3. Numerical method СSPH-TVD

also discussed.

It should be noted that the SWM is also actively applied and developed for theoretical research of various cosmic gas flows. The shallow water approximation allows describing a whole series of astrophysical objects: the protoplanetary and circumstellar disks [10, 11], the accretion disks around the compact relativistic objects [12], the cyclonic movements in the giant planets' atmospheres [13], and the spiral galaxies gas disk components [14]. The gravitational fields play the topography role in such astrophysical problems.

A lot of numerical methods have been proposed for the shallow water dynamics modeling employed for diverse tasks and conditions. Despite the fact that conservative methods of finite volume poorly describe stationary states, they allow correctly calculating shock waves and contact discontinuities. The latter problem could be overcome by the so-called well-balanced (WB) circuits [15–17].

Our main aim is through calculation for a flow with various Froude number within 0 ≤ Fr < 100 (Fr <sup>=</sup> <sup>u</sup>/cg, where cg <sup>¼</sup> ffiffiffiffiffiffi gH p is an analog of the sound speed, H is the depth, u is the flow velocity, g is the specific gravitational force) in order to simulate subcritical (Fr < < 1), transcritical (Fr~1), and supercritical (Fr > > 1) flows. Highly heterogeneous terrain topography including vertical discontinuities and small-scale inhomogeneities at the computational domain boundary makes the calculations noticeably complicated. The latter leads to special quality requirements in numerical algorithms. Hence, the modern numerical schemes should simulate fluid movements along the dry bottom and correctly describe the interfaces between wet and dry bottom [1, 18, 19].

Among the numerous numerical methods solving shallow water equations (SWEs), the following methods should be mentioned: the discontinuous Galerkin method based on triangulation [8], the weighted surface-depth gradient method for the MUSCL scheme [18], and the modified finite difference method [20]. The so-called constrained interpolation profile/multimoment finite-volume method utilizing the shallow water approximation is developed to simulate geophysical currents on a rotating planet in spherical coordinate system ([9] and see the references there). The particle-mesh method demonstrates good opportunities for calculation of rotating shallow water [21]. As a rule, numerical schemes of the second-order accuracy give satisfactory results and allow to correctly solve a wide range of tasks for diverse applications [17]. Special attention should be focused on the numerical way of a source term setting, since in the case of discontinuous topography, it may induce an error at the shock wave front [22].

We consider the original CSPH-TVD (combined smoothed particle hydrodynamics—total variation diminishing) algorithm of numerical integration of the Saint-Venant equations. It accounts for the new modifications improving the computational properties of the scheme. A detailed description of the numerical scheme is the main aim of the chapter.
