Curvilinear High-Order Mimetic Differences that Satisfy Conservation Laws

Abstract

We investigate the construction and usage of mimetic operators in curvilinear staggered grids. Specifically, we extend the Corbino-Castillo operators so they can be utilized to solve problems in non-trivial geometries. We prove that the resulting curvilinear operators satisfy the discrete analog of the extended Gauss-Divergence theorem. In addition, we demonstrate energy and mass conservation in curvilinear coordinates for the acoustic wave equation. These findings are illustrated in two-dimensional and three-dimensional elliptic/hyperbolic equations and can be extended to other partial differential equations as well.

Corrected typo on page 8, equation 11. Also on page 8, changed the sentence "where P and Q are diagonal weight matrix norms." to "where Q and P are diagonal positive definite matrices."