A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations

Abstract : The Serre–Green–Naghdi (SGN) equations, also known as the fully-nonlinear Boussinesq wave equations, accurately describe the behaviour of dispersive shoaling water waves. This article presents and validates a novel combination of methods for the numerical approximation of solutions to the SGN equations. The approach preserves the robustness of the original finite-volume Saint-Venant solver, in particular for the treatment of wetting/drying and equilibrium states. The linear system of coupled vector equations governing the dispersive SGN momentum sources is solved simply and efficiently using a generic multigrid solver. This approach generalises automatically to adaptive quadtree meshes. Adaptive mesh refinement is shown to provide orders-of-magnitude gains in speed and memory when applied to the dispersive propagation of waves during the Tohoku tsunami. The source code, test cases and examples are freely available.
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Journal of Computational Physics, Elsevier, 2015, 302, pp.336-358. 〈10.1016/j.jcp.2015.09.009〉
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Stéphane Popinet. A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations. Journal of Computational Physics, Elsevier, 2015, 302, pp.336-358. 〈10.1016/j.jcp.2015.09.009〉. 〈hal-01163101v2〉

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