Abstract:
Highresolution finite volume methods for
hyperbolic systems of PDEs are often based on
Godunov's method, with the addition of secondorder
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 wavepropagation 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 highresolution
correction terms. This approach has allowed the
development of generalpurpose software that applies
to a wide range of hyperbolic problems, including
problems that are not in conservation form and to
quasisteady 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.
