Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations.

*(English)*Zbl 1423.76220Summary: This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second-order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non-conforming linear elements for both velocities and elevation (\(P_1^{\text{NC}}-P_1^{\text{NC}}\)), is presented, giving optimal rates of convergence in all test cases. \(P_1^{\text{NC}}-P_{1}\) and \(P_1^{\text{DG}}-P_{1}\) mixed formulations lack convergence for inviscid flows. \(P_1^{\text{DG}}-P_{2}\) pair is more expensive but provides accurate results for all benchmarks. \(P_1^{\text{DG}}-P_1^{\text{DG}}\) provides an efficient option, except for inviscid Coriolis-dominated flows, where a small lack of convergence is observed.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

35Q35 | PDEs in connection with fluid mechanics |

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

##### Keywords:

finite element; shallow water equations; discontinuous Galerkin; non-conforming element; Riemann solver; convergence##### Software:

Gmsh
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\textit{R. Comblen} et al., Int. J. Numer. Methods Fluids 63, No. 6, 701--724 (2010; Zbl 1423.76220)

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