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**Godunov-type numerical scheme for the shallow water equations with horizontal temperature gradient.**
*(English)*
Zbl 1434.76082

Summary: We present a Godunov-type scheme for the shallow water equations with horizontal temperature gradient and variable topography. First, the exact solutions of the Riemann problem in a computational form are given, where algorithms for computing these solutions are described. Second, a Godunov-type scheme is constructed relying on exact solutions of the local Riemann problems. Computing algorithms for the scheme are given. The scheme is shown to be well-balanced and preserve the positivity of the water height. Numerical tests show that the scheme is convergent with a good accuracy, even for the resonant phenomenon, where the exact solutions contain several distinct waves propagating with the same shock speed. Furthermore, the scheme also provides us with good results for the solution of the wave interaction problem.

### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

### Keywords:

shallow water equations with temperature; topography; nonconservative; Godunov scheme; accuracy; resonant
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\textit{N. X. Thanh} et al., Taiwanese J. Math. 24, No. 1, 179--223 (2020; Zbl 1434.76082)

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