**A wave propagation algorithm for hyperbolic systems on curved
manifolds, **

by J. A. Rossmanith, D. S. Bale, and R. J. LeVeque
J. Comput. Phys. 199 (2004), pp. 631-662.

**Abstract.**
An extension of the wave propagation algorithm first introduced
by LeVeque [J. Comp. Phys. 131, 327--353 (1997)] is developed
for hyperbolic systems on a general curved manifold. This extension is
important in a variety of applications, including
the propagation of sound waves on a
curved surface, shallow water flow on the surface of the Earth,
shallow water magnetohydrodynamics in the solar tachocline,
and relativistic hydrodynamics in the presence of compact
objects such as neutron stars and black holes.
As is the case for the Cartesian wave propagation algorithm,
this new approach is second order accurate for smooth flows and
high-resolution shock-capturing. The algorithm is formulated such that
scalar variables are numerically conserved and vector
variables have a geometric source term that is naturally
incorporated into a modified Riemann solver. Furthermore,
all necessary one-dimensional Riemann problems are solved in a
locally valid orthonormal basis. This orthonormalization
allows one to solve Cartesian Riemann problems that are
devoid of geometric terms. The new method is tested
via application to the linear
wave equation on a curved manifold as well as the
shallow water equations on part of a sphere.
The proposed algorithm has been implemented in
the software package CLAWPACK
and is freely available on the web.

pdf file of preprint (Revised February, 2004.)

The CLAWMAN software implements these algorithms. Some sample simulations can be viewed on the claw/extensions/clawman webpages.

**bibtex entry:**

@Article{db-rjl-jr:manifolds,

author = "J. A. Rossmanith and D. S. Bale and R. J. LeVeque",

title = "A wave propagation algorithm for hyperbolic systems on

curved manifolds",

journal = "J. Comput. Phys.",

volume = "199",

year = "2004",

pages = "631--662",

URL = "http://www.sciencedirect.com/science/journal/00219991",

}