Source-linked AI summary
The GeoClaw software for depth-averaged flows with adaptive refinement
Marsha J. Berger, David L. George, Randall J. LeVeque, Kyle Mandli
TL;DR
Large-scale geophysical flows require efficient depth-averaged modeling that resolves evolving shorelines and realistic domains. GeoClaw combines high-resolution finite volume methods, adaptive mesh refinement, dry-state handling, and spherical or Cartesian grids in an open-source software framework. The paper illustrates its use for tsunami and dam-break simulations, while identifying data, visualization, and application-scope limitations.
Problem
Large geophysical flow problems need efficient spatial resolution, while moving wet-dry interfaces, realistic topography, and steady states create modeling challenges.
Method
GeoClaw uses open-source high-resolution finite volume methods with adaptive mesh refinement, dry-state solvers, and tools for Cartesian or spherical geophysical simulations.
Results
GeoClaw is illustrated through tsunami and dam-break modeling, with finer 1.25′ resolution capturing the leading tsunami peak better and estimating the primary-wave amplitude well.
Takeaways & Limitations
The software provides a framework for efficiently modeling large-scale geophysical flows and can be extended to other depth-averaged equation systems.
Takeaways & Limitations
Tsunami modeling remains uncertain because seafloor displacement is only approximated, while three-dimensional plotting tools are still under development.
Abstract
from arXiv · showhide
Many geophysical flow or wave propagation problems can be modeled with two-dimensional depth-averaged equations, of which the shallow water equations are the simplest example. We describe the GeoClaw software that has been designed to solve problems of this nature, consisting of open source Fortran programs together with Python tools for the user interface and flow visualization. This software uses high-resolution shock-capturing finite volume methods on logically rectangular grids, including latitude--longitude grids on the sphere. Dry states are handled automatically to model inundation. The code incorporates adaptive mesh refinement to allow the efficient solution of large-scale geophysical problems. Examples are given illustrating its use for modeling tsunamis, dam break problems, and storm surge. Documentation and download information is available at www.clawpack.org/geoclaw
1. Introduction
GeoClaw addresses large-scale geophysical flow problems with depth-averaged models, adaptive refinement, and specialized treatment of realistic topography, dry states, and steady states. It extends Clawpack with open-source tools and supports applications including tsunamis and dam-break floods.
- Motivation: Large geophysical domains make detailed three-dimensional fluid modeling inefficient, while depth-averaged equations can provide sufficiently accurate models.Adaptive mesh refinement concentrates cells where the evolving flow requires resolution without refining distant or unreached regions.
- Software contribution: GeoClaw uses high-resolution finite volume methods with adaptive mesh refinement to solve geophysical flow problems.The software is open source and has been used with shallow water equations for tsunami and dam-break modeling.
- Software contribution: GeoClaw is based on Clawpack and incorporates modifications for depth-averaged equations over realistic topography or bathymetry.Its framework supports multiple topographic datasets, longitude-latitude grids, and extensions to other depth-averaged systems.
- Numerical challenges: Moving wet-dry interfaces require dynamically changing dry cells and sufficient shoreline resolution, including under adaptive refinement.Cells can switch between wet and dry states to represent evolving flows or inundation.
- Numerical challenges: Well-balanced methods are needed to preserve nontrivial steady states such as an ocean at rest while resolving small perturbations such as tsunamis.This requirement arises in finite volume methods that use fluid depth as a primary conserved variable.
2. Depth-averaged mathematical models
The paper formulates geophysical flows with two-dimensional depth-averaged shallow water equations whose conserved variables are depth and horizontal momentum. Topography, drag, Coriolis effects, wet-dry boundaries, and model-scope assumptions determine how these equations are applied.
- Model formulation: The shallow water equations are the simplest two-dimensional depth-averaged model for fluid flow.They describe depth and horizontal momentum over topography or bathymetry, with drag and optional Coriolis terms.
- Model formulation: The conserved state is q = (h, hu, hv), combining fluid depth with momentum in the two horizontal directions.In the absence of bathymetry and drag source terms, the equations conserve mass and horizontal momentum.
- Steady states: For an ocean at rest, h(x, t) + B(x, y) remains constant at sea level, requiring exact numerical cancellation of topography and pressure terms.Maintaining this balance is critical for simulations involving small perturbations around equilibrium.
- Source terms: GeoClaw uses a drag source term with a Manning coefficient of 0.025 in the reported experiments.Typical Manning coefficients are empirically based on the substrate.
- Model scope: The shallow water equations are generally appropriate for tsunamis because their wavelengths are long relative to ocean depth, although dispersive terms may sometimes be needed.For shallow flow on steep terrain, vertical acceleration terms may also improve the model.
3. Numerical methods
GeoClaw combines Godunov-type high-resolution finite volume methods, f-wave well balancing, transverse wave propagation, and dry-state solvers within adaptive-grid computations. Its mapped-grid treatment supports Cartesian and spherical longitude-latitude domains, with some current domain restrictions.
- Finite volume method: GeoClaw computes cell-average solutions on logically rectangular grids using approximate Riemann problems at cell interfaces.The Godunov-type method robustly handles discontinuities such as hydraulic jumps and is second-order accurate in smooth regions.
- Well balancing: The f-wave formulation incorporates topography source terms directly into interface Riemann problems and preserves the ocean-at-rest equilibrium.At equilibrium, the modified flux difference and resulting f-waves are zero, so the solution does not change.
- Wave propagation: Transverse Riemann problems propagate waves into adjacent rows, improving stability and accuracy alongside second-order correction terms.Wave limiters are applied before calculating these correction terms.
- Dry states: Dry-state Riemann solvers handle initially dry states, cells drying as waves recede, arbitrary topography jumps, and interfaces between AMR levels.This enables implicit moving-shoreline representation through wet and dry cells.
- Mapped grids: On spherical longitude-latitude grids, GeoClaw uses spherical distances and areas while solving Riemann problems normal to cell edges.Current domain choices are limited to spherical longitude-latitude or purely Cartesian domains because general quadrilateral topography integration is unfinished.
4. Adaptive mesh refinement
GeoClaw uses patch-based adaptive mesh refinement to resolve geophysical flows spanning large spatial scales while tracking evolving features. Its refinement procedures address stability, conservation, wet-dry interfaces, and tsunami-specific time-stepping needs.
- Refinement strategy: Patch-based refinement spans orders of magnitude in spatial scales, from ocean-scale propagation to meter-scale resolution near small coastline features.Multiple refinement levels can be used, with fine resolution restricted to a small subset of a very large domain.
- Time stepping: Refined patches use recursively synchronized time steps to maintain stability of the explicit finite volume method.When spatial refinement is a factor of 4, the standard procedure takes four fine-level time steps per coarser-level step.
- Time stepping: Anisotropic time refinement can use smaller temporal than spatial refinement factors because shallow-water wave speeds decrease over shallower coastal regions.For the cited depths, the temporal refinement factor could be up to 6 times smaller than the spatial factor, reducing computing time on finest grids.
- Refinement strategy: Cells are flagged using error estimates, solution gradients, tsunami surface-elevation thresholds, or user-specified rectangular regions.Flagged cells are clustered into efficient, properly nested refinement patches.
- Conservation and wet-dry interfaces: AMR regrids by creating new patches and interpolating from existing fine grids, while preserving sea level and mass through topography-aware procedures.Surface elevation h+B is interpolated before fine-cell depth is computed; removing grids uses volume-weighted averaging.
- Conservation and wet-dry interfaces: Wet-dry interfaces require special treatment because refinement can introduce nonzero water depth into fine cells representing a shoreline.A coarse shoreline cell may be dry while its refined cells include wet cells needed to represent the shoreline and maintain constant sea level.
5. Topography data sets
GeoClaw combines topography data sets at different resolutions and converts their pointwise values into consistent cell averages. This supports multiscale simulations but can introduce discontinuities where fine-resolution data sets meet coarser ones.
- Input data: Users provide one or more rectangular-grid topography files specifying terrain elevation relative to sea level.Some applications also require files describing topographic displacement over time, such as earthquake-generated seafloor motion.
- Input data: Coarse ocean-scale data sets can be sufficient because tsunami wavelengths typically range from tens to hundreds of kilometers.The cited 10-minute and 4-minute ETOPO2 resolutions are roughly 18.5 km and 7.5 km near the equator.
- Resolution selection: At each location, GeoClaw selects the finest-resolution topography data set covering that point, allowing arbitrary numbers of files in one run.The effective topography will generally be discontinuous along boundaries between fine-scale data sets.
- Cell averaging: GeoClaw constructs a piecewise bilinear representation of pointwise topography and integrates it exactly to obtain cell-average topography values.The implementation handles cells spanning multiple data sets and guarantees consistency across adaptive mesh refinement levels.
6. Software tools and user interface
GeoClaw combines Fortran computational libraries with Python-based configuration, visualization, and extensibility tools. The software supports AMR-aware plotting and allows users to add physical source terms, alternative Riemann solvers, and custom refinement criteria.
- Architecture and interface: Most core computation is performed by Fortran routines, while Python modules support input preparation, topography operations, and user-facing workflows.Python scripts write parameter files read by the stand-alone Fortran code at run time.
- Visualization: Python and Matlab tools provide visualization on adaptive or fixed grids, including gauge plots of depth versus time at fixed locations.Interactive and hardcopy workflows support frame browsing, zooming, and generated webpages.
- Visualization: Three-dimensional surface-plot development remained ongoing, with Mayavi and VisIt under investigation.VisIt is described as supporting large-scale visualization, AMR data, and distributed-memory supercomputers.
- Extensibility: GeoClaw can incorporate additional physics through modified Riemann solvers or source terms such as friction, Coriolis effects, and wind forcing.A fractional-step procedure alternates homogeneous-system and source-term steps, requiring a user-supplied source-term time-step routine.
- Extensibility: Users can modify refinement criteria, including alternatives based on fluid momentum or speed rather than only sea-level displacement and fixed regions.The current approach combines displacement from sea level with user-specified fixed refinement regions.
7. Applications
GeoClaw is illustrated through a synthetic tsunami test and a real 2010 Chile tsunami simulation, demonstrating adaptive refinement, spherical-grid propagation, and computational efficiency. The examples also expose uncertainty in tsunami source data and the method’s behavior near refinement and outflow boundaries.
- Applications: Three GeoClaw applications include a synthetic tsunami test, the 27 February 2010 Chile earthquake, and the Malpasset dam catastrophe.The examples cover synthetic validation, real data, and a benchmark dam-break problem.
- Synthetic tsunami test: The synthetic test places a Gaussian hump in a radially symmetric spherical ocean with a variable continental shelf and an island whose angular location changes between tests.The setup is designed to assess whether solutions with different island locations are rotations of one another.
- Synthetic tsunami test: Adaptive refinement resolves the spreading wave in refined regions while the coarse grid poorly resolves it, and the two island placements produce nearly identical shore gauge histories.The wave propagates as a circle on the sphere, though its longitude–latitude representation is not circular.
- Synthetic tsunami test: 55 million grid cells were advanced in each synthetic calculation, which required roughly 18 minutes on one processor.Approximately 14M cells were on level 3, 28M on level 4, and 11M on level 5; levels 1 and 2 combined accounted for less than 1M.
- The 27 February 2010 tsunami: Tsunami modeling remains limited by uncertainty in earthquake-generated seafloor displacement and the paucity of measurements needed for detailed source reconstruction.The Okada model approximates displacement, while DART and other measurements support source inversion but are sparse.
- The 27 February 2010 tsunami: In the Chile event, a 4-level simulation with 1.25′ finest resolution captured the leading peak better and well estimated the primary-wave amplitude.The 3-level results matched a uniform-grid run initially, but became less accurate after the buoy region was no longer refined.
- Numerical behavior: The simulations show little evidence of spurious reflected waves at refinement boundaries, while non-reflecting outflow behavior is important for domains that omit the full ocean.The Godunov-type wave-propagation algorithms use constant extrapolation into ghost cells for outflow boundaries.
- The 27 February 2010 tsunami: The 3-level Chile computation took about 1.5 minutes and advanced 21 million cells, versus about 8 minutes and 137 million cells for the comparable uniform grid.The authors describe the AMR advantage as clear even without coastal inundation zooming.
8. Conclusions and future plans
GeoClaw is a robust and stable open-source code for geophysical flow simulation, with ongoing development toward broader applications and improved computing. Its examples and tools support tsunami, dam-break, and related flow modeling.
- GeoClaw’s released version is described as fairly robust and stable, while development continues.The authors were incorporating OpenMP to use multi-core shared-memory computers.
- The software supports ongoing development of depth-averaged models for granular-fluid mixtures in debris flows, lahars, and mudslides.These applications occur in rugged mountainous regions and present difficulties similar to overland flooding.
- Animations and GeoClaw code accompanying the presented simulations are available through the project’s webpage and application gallery.