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Stabilized residual distribution for shallow water simulations. (English) Zbl 1330.76097
Summary: We propose a stabilized Residual Distribution (\(\mathcal{RD}\)) scheme for the simulation of shallow water flows. The final discretization is obtained combining the stabilized \(\mathcal {RD}\) approach proposed in (Abgrall, J. Comp. Phys. 214, 2006) and (Ricchiuto and Abgrall, \(ICCFD4\), Springer-Verlag 2006), with the conservative formulation already used in (Ricchiuto et al., J. Comp. Phys. 222, 2007) to simulate shallow water flows. The scheme proposed is a nonlinear variant of a Lax-Friedrichs type discretization. It is well balanced, it actually yields second-order of accuracy in smooth areas, and it preserves the positivity of the height of the water in presence of dry areas. This is made possible by the residual character of the discretization, by properly adapting the stabilization operators proposed in (Abgrall, J. Comp. Phys. 214, 2006) and (Ricchiuto and Abgrall, \(ICCFD4\), Springer-Verlag, 2006), and thanks to the positivity preserving character of the underlying Lax-Friedrichs scheme. We demonstrate the properties of the discretization proposed on a wide variety of tests.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M12 Finite volume methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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