Workshop on Foundations of Numerical PDEs

July 7 - July 9, 2005

Universidad de Cantabria in Santander, Spain

High-Resolution Finite Volume Methods and Approximate Riemann Solvers for Hyperbolic Systems

Randall LeVeque

Department of Applied Mathematics at University of Washington

Abstract:   High-resolution finite volume methods for hyperbolic systems of PDEs are often based on Godunov's method, with the addition of second-order correction terms that are limited in some manner to avoid nonphysical oscillations and sharpen the resolution of steep gradients or discontinuities such as shock waves. This is often coupled with upwinding based on a characteristic decomposition or the solution of the Riemann problem between the neighboring states. A variety of slope limiting or flux limiting methods have been proposed and successfully used. The wave-propagation method is a general formulation of such methods that limits the waves resulting from the Riemann solution and uses the resulting waves to define the high-resolution correction terms. This approach has allowed the development of general-purpose software that applies to a wide range of hyperbolic problems, including problems that are not in conservation form and to quasi-steady balance laws where the flux gradient is nearly balanced by a source term. In practice approximate Riemann solvers are often used, ranging from very simple approximations such as the HLL solver to more sophisticated and expensive solvers that better mimic important aspects of the exact solution. Several of these can be unified by interpreting them in the framework of "relaxation Riemann solvers" that are related to relaxation schemes for hyperbolic systems.