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Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations. (English) Zbl 1423.76220
Summary: 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
Software:
Gmsh
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