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Implicit Adaptive Mesh Refinement for Dispersive Tsunami Propagation: Algorithm and Computational Examples

Core Concepts
Implicit adaptive mesh refinement enhances accuracy in modeling dispersive tsunami propagation.
The content introduces an algorithm for solving the dispersive depth-averaged Serre-Green-Naghdi equations using patch-based adaptive mesh refinement. The method is implicit, allowing different time steps on various refinement levels. Computational examples demonstrate stability and accuracy in realistic tsunami modeling scenarios, including a hypothetical asteroid impact generating a short-wavelength tsunami. The implementation is part of the GeoClaw software widely used for hazard modeling. Introduction: Algorithm to solve dispersive SGN equations with adaptive mesh refinement. Implicit method requiring elliptic system solution at each time step. Widely used GeoClaw software enhanced for shorter wavelength phenomena. Equations: Nonlinear shallow water equations (SWE) vs. Boussinesq-type equations. Depth-averaged system crucial for trans-oceanic propagation. Comparison between SWE and dispersive models based on length scales. Solution Algorithm: Splitting method with elliptic equation solution and shallow water step. Patch-based AMR strategy with subcycling in time. Composite solve without subcycling for comparison. Computational Examples: Radially symmetric ocean, shelf, and beach scenario tested for accuracy and stability. Results show dispersion effects, soliton fission, wave breaking near shore. Comparison of coupled vs uncoupled algorithms and uniform grid solutions. Convergence analysis with 4, 5, and 6 level adaptive runs.
この方程式は、水深が水平長さスケールよりもはるかに小さいことを前提とした長波近似である。 これらの方程式は、非分散であり、線形化された方程式の分散関係は一定の波速を示す。 SGNシステムの方程式は、追加の源項を持つSWE(2.1)の形をしている。
"Several figures in this paper show outlines of the refined patches." "Our implementation... can still solve large-scale realistic tsunami modeling problems on a laptop."

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