We present a numerical method for the accurate
and efficient solution of hyperbolic conservation
laws on curved manifolds. Such equations arise
in several important applications including
the propagation of sound waves on a
curved surface, large-scale dynamics of the atmosphere,
and relativistic hydrodynamics in the presence of compact
objects such as black holes. The proposed method 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 advection and shallow water equations on the surface of a sphere.
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