Randy LeVeque: Finite-Volume Methods and Software for Hyperbolic PDEs and Conservation Laws
2005 Harvey Mudd College Mathematics Conference on Scientific Computing
Randy LeVeque (University of Washington, Seattle)
Hyperbolic systems of partial differential equations frequently arise when modeling phenomena involving wave propagation or advective flow. Finite-volume methods are a natural approach for conservation laws since they are based directly on integral formulations and are applicable to problems involving shock waves and other discontinuities. High-resolution shock-capturing methods developed originally for compressible gas dynamics can also be applied to other hyperbolic systems, even if not in conservation form. I will describe a robust class of wave-propagation methods that have been implemented in the CLAWPACK (Conservation LAW PACKage) software for solving hyperbolic problems in one, two, and three space dimensions. Adaptive mesh refinement capabilities are also included. This software has been applied to a variety of problems in diverse fields, including gas dynamics, multiphase flows, linear and nonlinear elasticity, combustion, biological flows, and numerical relativity. Some examples will be shown from recent work on geophysical flows modeling volcanoes and tsunamis.