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The generalized Riemann problem method for the shallow water equations with bottom topography. (English) Zbl 1178.76249
Summary: This paper extends the generalized Riemann problem method (GRP) to the system of shallow water equations with bottom topography. The main contribution is that the generalized Riemann problem method [M. Ben Artzi and J. Falcovitz, J. Comput. Phys. 55, 1–32 (1984; Zbl 0535.76070)] is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in the numerical fluxes, and at the same step the source term is discretized with an interface method in which only mid-point values are plugged in. This scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method (SGM) [J. G. Zhou et al., J. Comput. Phys. 168, No. 1, 1–25 (2001; Zbl 1074.86500)] into the data reconstruction step, and it is also suitable for both steady and unsteady flow simulations. We illustrate the accuracy of this scheme by several 1D and 2D numerical experiments.
MSC:
76M12Finite volume methods (fluid mechanics)
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction