4. The set of test problems verifying the numerical models

In the paragraph, the basic set of various tests for one- and two-dimensional problems suitable for revelation of positive and negative sides of the numerical shallow water models is reviewed. According to Toro [26], there are four main test types appropriate to evaluate the numerical solutions (Figure 5).

	- a. Dam-break waves propagating on the wet bed [16, 36] with the initial distribution of the water depth H(x ≤ 0) = H<sup>01</sup> and H(x > 0) = H02, (H<sup>01</sup> > 0, H<sup>02</sup> > 0). This solution is an analog of the pressure jump decay in an ordinary hydrodynamics. It also contains a hydraulic shock (an analog of a shock wave) and a rarefaction wave for which the exact solutions of the RP for a self-similar type f(x, t) = f(ξ) f(x/t) exist [37]. The bora (hydraulic jump) front for the CSPH-TVD scheme is smeared into three cells and does not contain dispersion oscillations typical for sufficiently good numerical schemes.
	- b. Dam-break waves propagating on the dry bed [16]: H(x ≤ 0) = H<sup>01</sup> and H(x > 0) = 0. This is an important test imposing special requirements on a numerical scheme for a

Figure 5. The bottom profiles with known exact solutions of SWE.

correct description of the moving boundary between the water and dry bottom. It also allows comparison of the numerical result with exact solution of the RP [37].


The CSPH-TVD method demonstrates a good ability for parallelization on graphics processors [35]. Figure 4 shows the computer implementation of the CSPH-TVD method on GPUs. It

Figure 4. The flow diagram for each of the calculation module: K1 determines the presence of water in the CUDA block; K2 calculates the forces at the time moment tn at the Lagrangian's stage; K3 calculates the time step tn + 1; K4 calculates the new positions of the particles and their integral characteristics at time tn + 1/2; K5 defines the forces on the time layer tn + 1/2; K6 calculates the positions of the particles and their integrated characteristics for the next time layer tn + 1; K7 calculates the flux of physical quantities through the cells boundaries at time moment tn + 1/2; and K8 determines the final hydrodynamic

In the paragraph, the basic set of various tests for one- and two-dimensional problems suitable for revelation of positive and negative sides of the numerical shallow water models is reviewed. According to Toro [26], there are four main test types appropriate to evaluate the numerical

a. Dam-break waves propagating on the wet bed [16, 36] with the initial distribution of the water depth H(x ≤ 0) = H<sup>01</sup> and H(x > 0) = H02, (H<sup>01</sup> > 0, H<sup>02</sup> > 0). This solution is an analog of the pressure jump decay in an ordinary hydrodynamics. It also contains a hydraulic shock (an analog of a shock wave) and a rarefaction wave for which the exact solutions of the RP for a self-similar type f(x, t) = f(ξ) f(x/t) exist [37]. The bora (hydraulic jump) front for the CSPH-TVD scheme is smeared into three cells and does not contain dispersion oscillations typical for sufficiently good numerical schemes.

b. Dam-break waves propagating on the dry bed [16]: H(x ≤ 0) = H<sup>01</sup> and H(x > 0) = 0. This is an important test imposing special requirements on a numerical scheme for a

indicates the execution sequence of CUDA kernels at various stages of our algorithm.

4. The set of test problems verifying the numerical models

parameters at time moment tn + 1 (see details in Ref. [35]).

246 Numerical Simulations in Engineering and Science

1. The comparison with the exact solutions for several 1D problems.

Figure 5. The bottom profiles with known exact solutions of SWE.

solutions (Figure 5).

	- a. Dam-break flow over the initially dry bed with a bottom obstacle is a tough test for a simple SWM due to the formation of negative wave propagation [42].

b. The experimental results of the dam-break on a dry bed channel with varying width are used to examine numerical models [16]. The laboratory measurements of flow parameters for the dam-break in the nonprismatic converging-diverging channel are described in Ref. [38].

of the instability development of the tangential velocity discontinuity in shallow water (see Figure 6a) are obvious: (I) the formation of the eigenmode; (II) the linear stage of instability development, when perturbations increase according to the law ∝ exp(t/T)(T ≈ 9.6 for Fr = 1); (III) the nonlinear mode of perturbations evolution, when the growth of the perturbation amplitude ceases and the characteristic vortex structures are formed (so-called cat eyes, Figure 6b). Thus, in the CSPH-TVD method, there is a lack of important disadvantage of classical SPH

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

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We have developed software to solve some engineering applications utilizing the parallel

1. The hydrological regimes of the spring flooding in the Volga-Akhtuba interfluve have been studied thoroughly [1, 35]. The optimal hydrographs of the Volga Hydroelectric Power Station have been constructed. And finally a new approach optimizing the hydrotechnical

2. Our numerical experiments have reproduced the dynamics of the catastrophic flooding wave in the city of Krymsk area (Russia, Caucasus Mountains) in July 2012, leading to massive losses of life. A number of features of the hydrological regime during the flash flood of 2012 [49], associated with the landscape and the distribution of rainfall has been

3. Figure 7 shows the more detailed results of the numerical simulations of the tsunami formation in the Pacific Ocean during an earthquake off the coast of Japan in 2011. A bilateral hydraulic shock (tsunami) is formed due to the displacement of tectonic plates and then the waves propagate to both sides of the discontinuity. The tsunami reaches the shores of Japan in 20–40 min after the earthquake, while the wave height Hw reaches

We have described in detail the numerical method for solving the equations of hydrodynamics in the approximation of the SWM. Our method is a hybrid scheme successfully combining positive properties of Euler's and Lagrange's approaches. The CSPH-TVD method allows to model nonstationary flows on a complex heterogeneous bottom relief, containing kinks and sharp jumps of the bottom profile. The numerical scheme is conservative, well balanced and provides a stable through calculation in the presence of non-stationary "waterdry bottom" boundaries on the irregular bottom relief, including the transition through the

algorithm, which does not allow correctly modeling the tangential discontinuity.

5. Engineering applications

revealed.

6. Conclusion

computing boundary.

10–20 m at the coast line.

implementation of the CSPH-TVD method on GPUs.

projects has been developed [46–48].


Figure 6 shows the results of the numerical simulations of shear flow instability (Kelvin-Helmholtz instability) in shallow water using the CSPH-TVD method. The characteristic stages

Figure 6. The problem of the tangential velocity discontinuity is represented by: (a) the time dependence of the minimum (shown by solid line) and maximum (shown by dashed line) of the amplitude of the water depth disturbances Hw(t) = H (t) H<sup>0</sup> (H<sup>0</sup> H(0)); (b) the spatial distribution of the vorticity of the velocity vector field ω = ∇ v/2 at t = 110. The dashed line shown the position of the tangential discontinuity.

of the instability development of the tangential velocity discontinuity in shallow water (see Figure 6a) are obvious: (I) the formation of the eigenmode; (II) the linear stage of instability development, when perturbations increase according to the law ∝ exp(t/T)(T ≈ 9.6 for Fr = 1); (III) the nonlinear mode of perturbations evolution, when the growth of the perturbation amplitude ceases and the characteristic vortex structures are formed (so-called cat eyes, Figure 6b). Thus, in the CSPH-TVD method, there is a lack of important disadvantage of classical SPH algorithm, which does not allow correctly modeling the tangential discontinuity.
