Implicit Adaptive Mesh Refinement for Dispersive Tsunami Propagation

TL;DR

The paper develops an implicit, patch-based AMR solver for the dispersive Serre-Green-Naghdi (SGN) equations within GeoClaw, enabling level-wise time subcycling by solving an elliptic system at each time step. The key idea is to update dispersive corrections through an elliptic solve $(I+\alpha\mathcal{T})\boldsymbol{\psi}=\mathbf{b}$, embed the resulting variables into the AMR state, and advance the hyperbolic SWE on each level with subcycled time steps. A composite, non-subcycled solver provides a baseline for accuracy comparisons, while sample problems—radial symmetry, the 2011 Japan tsunami, and a hypothetical asteroid impact—demonstrate stability, accuracy, and practicality on a laptop. The results show that the SGN-AMR approach can resolve short-wavelength dispersive waves with complex coastal topography, offering a scalable, high-fidelity alternative to non-dispersive models for hazards analysis and planetary defense studies. The work lays the groundwork for tighter nearshore SGN-SWE transitions, reduced inter-level reflections, and comparisons with hyperbolic dispersion formulations, with code and data openly available to the community.

Abstract

We present an algorithm to solve the dispersive depth-averaged Serre-Green-Naghdi (SGN) equations using patch-based adaptive mesh refinement. These equations require adding additional higher derivative terms to the nonlinear shallow water equations. This has been implemented as a new component of the open source GeoClaw software that is widely used for modeling tsunamis, storm surge, and related hazards, improving its accuracy on shorter wavelength phenomena. We use a formulation that requires solving an elliptic system of equations at each time step, making the method implicit. The adaptive algorithm allows different time steps on different refinement levels, and solves the implicit equations level by level. Computational examples are presented to illustrate the stability and accuracy on a radially symmetric test case and two realistic tsunami modeling problems, including a hypothetical asteroid impact creating a short wavelength tsunami for which dispersive terms are necessary.

Figures (15)

• Figure 1: \newlabelfig:gpvel0 The scaled group velocity $\omega'(k)/\sqrt{gh_0}$ for the dispersion relations shown in the text, plotted versus $2\pi/(kh_0)$, the wavelength relative to the undisturbed fluid depth. In this work we implement SGN with $\alpha=1.153$.
• Figure 1: There is a coarse grid with cells outlined in black on the base grid. Over part of the domain there are two adjacent fine patches refined by a factor of two, with cells outlined in red. The fine grid time step is half the coarse grid step. Before the fine grid takes a step, two ghost cells (not shown in the figure) are needed to complete the stencil. Figure taken from ICM22. \newlabelfig:amrfig0
• Figure 1: \newlabelfig:radocean_topo0 Topography for the radially symmetric ocean test case.
• Figure 2: Sample enumeration of cells on level 2 grids. The numbers are used to map to the row in the linear system. \newlabelfig:marking0
• Figure 2: \newlabelfig:radocean10 Solution to the radially symmetric ocean test problem at several times. The left column shows a colormap of surface elevation $\eta(x,y,t)$ over the positive quadrant. Colored rectangles show grid patches (level 2: magenta, level 3: blue, level 4: yellow, level 5: black) and the dashed line is the $x=y$ diagonal. The right column shows a transect of the 2D solution along the diagonal at each time, along with the solution to the 1D radially symmetric equations at the same time. Plots are zoomed in near the shore at time $t=1800$ seconds.