# Parallel, adaptive Cartesian grid algorithms for natural hazards modeling

**Agency :** National Science Foundation (NSF)

**Type :** NSF-DMS: #1819257 (single PI)

**Amount to BSU** : $315K

### Project summary

Geophysical hazards such as flooding, tsunamis, debris flows, landslides, storm surges and potential dam failures threaten communities across the United States and globally. This project develops computational tools that can efficiently simulate these hazards, enabling a diverse group of researchers and emergency planners to develop hazard maps of areas most likely to be impacted by these disasters. For efficiency, the computational framework uses adaptive, "depth-averaged" mathematical models that only require two dimensional planar grids, rather than fully three dimensional meshes. A primary goal of the project is to correct our depth-averaged model to capture localized waves that may spill over flood barriers or overtop harbor breakwaters. These correction terms will give current users of the computational framework critical additional capabilities for modeling shallow geophysical hazards and allow them to create more robust hazard maps. The computational tools can also take full advantage of emerging hardware trends available on desktop workstations, moderate sized compute clusters, as well as massively parallel computing facilities available at NSF funded supercomputing sites. The project also provides users with tools for visualizing results using open source software such as the Google Earth browser. Ultimately, computational modeling can aid responders in predicting how to distribute emergency resources in the event of unavoidable hazards and serve to inform developers, legislative representatives, and citizenry of potential risks in their communities.

The research will focus on the implementation of a direct solver for variable coefficient elliptic problems on adaptively refined quad-tree meshes. The targeted solver is the Hierarchical Poincare Steklov (HPS) solver, developed by A. Gillman and P. Martinsson. Satisfying four crucial properties, this solver (1) has the ease of use of matrix-free methods, (2) can solve nearby systems quickly, (3) has optimal O(N) efficiency, and (4) provides parameters that can be tuned to reduce computational cost in proportion to accuracy requirements. Furthermore, the method uses low rank approximations to compress dense matrices and accelerate matrix computations. In the proposed work, the PI will modify the original HPS solver for use with second order, finite volume schemes and implement the solver in ForestClaw, the parallel, patch-based Cartesian adaptive quad-tree code. The PI will also report on the scalability and parallel efficiency of the implementation of the HPS method. Two technical challenges that will arise are to develop effective procedures for merging Dirichlet-to-Neumann maps across processor boundaries and incrementally updating the solver factorization for dynamically evolving meshes. The targeted application is the solution to the Serre-Green Naghdi equations for modeling dispersive corrections to the shallow water wave equations. These corrections will be included in the GeoClaw extension of ForestClaw. GeoClaw (D. George, R. J. LeVeque, M. J. Berger) is a widely used software package for solving depth-averaged flow equations. The addition of these correction terms to the GeoClaw extension will provide GeoClaw users with critical capabilities for modeling tsunamis, flooding, debris flows, storm surges and other shallow geophysical flows. Ultimately, the proposed solver can be used within the ForestClaw framework as a general purpose elliptic solver for a variety of physical flow phenomena.

**PI:** Donna Calhoun, Boise State University

**Funding period : **6/2018-5/2023