Curvilinear High-Order Mimetic Differences that Satisfy Conservation Laws

TL;DR

This paper develops and proves the effectiveness of mimetic difference operators on curvilinear grids, ensuring conservation laws and energy/mass conservation for complex geometries in PDEs.

Contribution

It extends Corbino-Castillo operators to curvilinear grids, proving their conservation properties and demonstrating their application to various PDEs in complex geometries.

Findings

Operators satisfy discrete Gauss-Divergence theorem

Energy and mass conserved in curvilinear coordinates

Applicable to 2D and 3D PDEs

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.