Application of Discrete Exterior Calculus Method to the Heat Transport Equation in Porous Aquifers

Abstract

The numerical analysis of partial differential equations (PDEs) is a classic and highly active investigation area. In this area, there is a wide variety of established methods such as finite differences, finite elements, finite volume, spectral, etc. Furthermore, we can affirm that all of them have an analytical origin. In recent decades, geometric methods based on Exterior Calculus have been proposed. This is due to the geometric content of many Physics theories. In this group, we can highlight the Finite Exterior Element Calculus, or FEEC, a theoretical approach to designing and understanding finite element methods for the numerical solution of various PDEs. This method is substantiated by the traditional finite element functional analysis techniques, but it is accompanied by topology and homological algebra tools. An alternative method in this field is Discrete Exterior Calculus (DEC). Exterior Calculus generalizes vector calculus to high dimensions and differential manifolds. Discrete Exterior Calculus (DEC) is one of their discretizations, producing a numerical method for solving PDEs on simplicial complexes. In DEC, geometric operators on simplicial complexes are used in any dimension, and equivalent discrete versions are proposed for objects and differential operators, such as differential forms, vector fields, etc. DEC is proposed as a method for solving partial differential equations that consider the geometric and analytical characteristics of the operators and the domains over which that applies. Mathematical heat transfer models in porous media have recently received considerable attention. Second-order partial differential equations give these for heat and flow energy conservation. To study the thermal characteristics of conduction and advection within porous media, thermal equilibrium, and non-thermal equilibrium models. This work analyzes 2D numerical models of heat transport in porous aquifers with DEC.