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

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 equati...

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Bibliographic Details
Published in:Journal of computational physics 2015-12, Vol.302, p.336-358
Main Author: Popinet, Stéphane
Format: Article
Language:English
Subjects:
Adaptive mesh refinement
Boussinesq equations
Dispersion (wave)
Dispersive wave model
Fluid mechanics
Mathematical analysis
Mathematical models
Mechanics
Multigrid elliptic solver
Ocean, Atmosphere
Physics
Preserves
Quadtree
Sciences of the Universe
Solvers
Tsunami
Wave equations
Wave propagation
Well-balanced
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2015.09.009