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A geometrically intrinsic lagrangian-Eulerian scheme for 2D shallow water equations with variable topography and discontinuous data

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  • Abreu, Eduardo
  • Bachini, Elena
  • Pérez, John
  • Putti, Mario

Abstract

We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. This work was dictated by the fact that geometrically Intrinsic Shallow Water Equations (ISWE) are characterized by non-autonomous fluxes. Handling of non-autonomous fluxes is an open question for schemes based on Riemann solvers (exact or approximate). Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value problem that describes the evolution of the balance laws governing the geometrically intrinsic shallow water equations. The evolved solution set is then projected back to the original surface grid to complete the proposed Lagrangian-Eulerian formulation. The resulting scheme maintains monotonicity and captures shocks without providing excessive numerical dissipation also in the presence of non-autonomous fluxes such as those arising from the geometrically intrinsic shallow water equation on variable topographies. We provide a representative set of numerical examples to illustrate the accuracy and robustness of the proposed Lagrangian-Eulerian formulation for two-dimensional surfaces with general curvatures and discontinuous initial conditions.

Suggested Citation

  • Abreu, Eduardo & Bachini, Elena & Pérez, John & Putti, Mario, 2023. "A geometrically intrinsic lagrangian-Eulerian scheme for 2D shallow water equations with variable topography and discontinuous data," Applied Mathematics and Computation, Elsevier, vol. 443(C).
    • Handle: RePEc:eee:apmaco:v:443:y:2023:i:c:s009630032200844x
    DOI: 10.1016/j.amc.2022.127776
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    References listed on IDEAS

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    1. Qiao, Dianliang & Lin, Zhiyang & Guo, Mingmin & Yang, Xiaoxia & Li, Xiaoyang & Zhang, Peng & Zhang, Xiaoning, 2022. "Riemann solvers of a conserved high-order traffic flow model with discontinuous fluxes," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    2. Carlino, Michele Giuliano & Gaburro, Elena, 2023. "Well balanced finite volume schemes for shallow water equations on manifolds," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    3. Abreu, E. & Lambert, W. & Perez, J. & Santo, A., 2017. "A new finite volume approach for transport models and related applications with balancing source terms," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 137(C), pages 2-28.
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