Structure-preserving numerical discretizations for domains with boundaries

Abstract

This SAND report documents Exploratory Express LDRD Project 223790, "Structure-preserving numerical discretizations for domains with boundaries", which developed a method to incorporate consistent treatment of domain boundaries and arbitrary boundary conditions in discrete exterior calculus (DEC) for arbitrary polygonal (2D) and tensor-product structure prism (3D) grids. The new DEC required the development of novel discrete exterior derivatives, boundary operators, wedge products and Hodge stars. This was accomplished through the use of boundary extension and the blending of known 2D operators on the interior with 1D operators on the boundary. The Hodge star was based on the Voronoi Hodge star, and retained the limitation of a triangular circumcentric primal or dual grid along with low-order accuracy. In addition to the new DEC, two related software packages were written: one for the study of DEC operators on arbitrary polygonal and polyhedral grids using both symbolic and numerical approaches and one for a (thermal) shallow water testbed using TRiSK-type numerics. Immediately relevant (already funded, through CANGA) followup work is the development of a high-order, geometrically flexible Hodge star and structure-preserving, high-order, oscillation-limiting transport operators (using WENO) for n-forms on arbitrary 2D and 3D grids. This will provide all of the machinery required for a high-order version of TRiSK with boundaries on arbitrary 2D and tensor-product 3D grids, which is applicable to both the atmospheric (CRM in E3SM-MMF) and oceanic (MPAS-O) components of E3SM.