A wave propagation algorithm for hyperbolic systems on curved manifolds

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 shockcapturing. 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.

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

“…On a cubed-sphere grid, a spherical quadrilateral grid, the following works [37,33,31,45,35] achieved a two-dimensional momentum representation avoiding any pole problem. Our new model achieves the same flexibility but on unstructured triangular grids using spherical triangular coordinates.…”

“…All of these models rely directly on either the triangular grid being derived from the icosahedron or on a linear representation of the discrete operators; this way there is an easily computable dual grid which is based on hexagons (icosahedron) or the operators can be constructed using the vertices of an element which are co-planar (linear representation). However, for high-order operators on general triangular grids, one needs to construct the discrete spatial operators directly on the curved manifold which then requires the derivation of the Christoffel symbols from differential geometry (see [35] for a summary of the use of differential geometry for atmospheric flow).…”

Relativistic Bondi-Hoyle-Lyttleton accretion: A parametric study

“…We therefore had to transform the fluid variables to ensure that they were expressed with the correct metric. We note that the approach of Rossmanith et al (2004) is the rigorous way to deal with this, but that it would seem to be prohibitively computationally expensive. Notes.…”

General relativistic hydrodynamics on overlapping curvilinear grids

“…However, since the SLIC scheme requires metric values at cell faces, we need to describe how to determine these from cell-centred values. A careful and rigorous approach to this is presented in Rossmanith et al (2004). However, as this approach is somewhat involved, and computationally expensive, we instead use the following approach, which also includes considerations for a curvilinear coordinate system.…”