6. Conclusion

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 computing boundary.

Author details

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Alexander Khoperskov\* and Sergey Khrapov

Volgograd State University, Volgograd, Russia

\*Address all correspondence to: khoperskov@volsu.ru

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Figure 7. Numerical modeling of the tsunami occurred in 2011 off the coast of Japan: (a) the digital terrain model, where the size of the simulation area is 1500 1500 cells; the yellow line shows the area of tsunami formation; (b) the 3D wave structure 19 min after the earthquake; frames (c) and (d) are the tsunamis at different moments of time.

On the SPH stage, various smoothing cores can be applied, as well as various TVD-delimiters and methods for the RP solution depending on the features of the problem being solved. The scheme has the second order of convergence on smooth solutions and the first order on discontinuities that corresponds to the accuracy of the Godunov-type schemes. In the case of a non-uniform topography the CSPH-TVD method requires less computational resources than the Godunov's type schemes, when the WB-condition is necessary to fulfill for the numerical scheme. Comparing with various SPH-method modifications, the numerical CSPH-TVD scheme has higher accuracy and computational speed for the same number of particles; it is less dissipative and better balanced.
