Finite Volume Methods and Software for Hyperbolic Problems

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This proposal will fund research on finite volume methods for solving linear and nonlinear hyperbolic systems of partial differential equations, and the continued development of the open source software package Clawpack. This software provides an implementation of a class of wave propagation algorithms that has been extensively tested and used by students and researchers in many fields, and also incorporates adaptive mesh refinement (AMR). Specific goals of this research include: (1) Development of Discontinuous Galerkin (DG) methods in the Clawpack framework, which could potentially provide higher order of accuracy for a certain class of problems and greater efficiency in spite of their higher cost per time step. (2) Solution of three-dimensional orthotropic poroelastic wave propagation problems, with applications to the continuing investigation of Extracorporeal Shock Wave Therapy (ESWT). (3) Development of adjoint equation technology for AMR error estimation and sensitivity analysis, with application to tsunami modeling in particular. (4) Development of multilevel Monte-Carlo capabilities and exploration of this and other uncertainty quantification techniques, with application to the probabilistic assessment of natural hazards.The class of algorithms being investigated can be used to solve a wide range of practical problems in science and engineering that involve wave motion. These problems are modeled by partial differential equations that cannot be solved exactly, and so numerical approximations must be generated computationally at millions of grid points. Development of better algorithms for efficiently computing accurate solutions is a primary goal of this research. Much of the work is quite general and can be applied to many different problems, and the algorithms are implemented in an open source software package Clawpack that is freely available at www.clawpack.org. Work supported by this grant is strongly motivated by two specific applications of practical interest. One is the simulation of hazardous geophysical flows, particularly tsunamis, and the development of techniques to more rapidly and accurately predict the effects of a tsunami. For risk assessment and hazard mitigation, it is also necessary to consider a range of potential earthquakes that could cause tsunamis in the future. Mathematical and computational techniques will be investigated that can aid in efficiently estimating the probability of inundation from a large number of potential tsunamis at many different points in a coastal community. The other motivating application is the study of shock wave propagation in tissue and bone and the mechanical stress generated that can lead to biological response, such as bone growth or wound healing. These studies are important in gaining a better understanding of shock wave therapy, a clinical procedure that has exhibited some significant results.