Finite volume methods for solving hyperbolic partial differential equations on curved manifolds

Abstract

The natural mathematical arena to formulate conservation laws on curve manifolds is that of differential geometry. Ricci developed this branch of mathematics from 1887 to 1896. Subsequent work in differential geometry has made it an indespensible tool for solving in mathematical physics. The idea from differential geometry is to formulate hyperbolic conservation laws of scalar field equation on curved manifolds. The finite volume method 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. The 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.