Discrete calculus methods for diffusion

Abstract

A general methodology for the solution of partial differential equations is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations. The fact that the calculus is exact gives these methods the ability to capture the physics of PDE systems well. The construction of both node and cell based methods of first and second-order are described for the problem of unsteady heat conduction – though the method is applicable to any PDE system. The performance of these new methods are compared to classic solution methods on unstructured 2D and 3D meshes for a variety of simple and complex test cases.

Introduction

This paper is dedicated to Pieter Wesseling. It bears his hallmark at many levels. Philosophically, it is a paper about the intimate connection between physics and mathematics. Prof. Wesseling, an Aerospace Engineer turned Mathematician has always produced papers that are always keenly aware of the connection. Topically, it is a paper about staggered mesh methods – one of many areas in which Pieter and his coworkers are prolific (see the references in [1], [2]). And in particular, the paper addresses fundamental questions about how to apply staggered mesh methods to compressible flow problems – an area Pieter is particularly interested in [3], [4], [5], [6].

Staggered mesh methods have traditionally been applied to incompressible flows. The lack of pressure modes is particularly attractive in that application. There is therefore considerable literature addressing the issue of how to discretize the momentum equations with structured [7], curvilinear [8], [9], and unstructured staggered mesh methods [10], [11], [12], [13], [14], [30], [31]. However, in the context of compressible flow there arises the additional issue of how to discretize the density and energy equations.

The discrete differential operators in incompressible staggered mesh methods have very unique and attractive mathematical properties that allow the discrete equations to physically mimic their continuous counterparts. This not only leads to a lack of pressure modes, but kinetic energy and vorticity conservation statements [15], [16], maximum principles and many other attractive properties [17]. If we wish the compressible discretization to also have these sorts of attractive physical properties (like entropy increase), then presumably the scalar equations (density and energy) must also be discretized appropriately.

Until recently, it was not clear to the authors what criteria should be used to judge if a scalar transport equation was discretized ‘appropriately’. We believe this dilemma is addressed by the Discrete Calculus approach presented in this paper. In order to carefully explain the Discrete Calculus approach, this paper actually only focuses on the unsteady diffusion equation (not the advection–diffusion equation). The diffusion term contains sufficient complexity to present the fundamental ideas of the Discrete Calculus approach. Due to space limitations, the issues concerning advection must be addressed in a subsequent paper.

The premise of this paper is that numerical methods that capture the physics of the equations well have an associated exact Discrete Calculus. The fact that PDE’s can always be discretized exactly is demonstrated in the Section 2. To make the presentation clear and concrete the paper focuses on the diffusion (or heat) equation. However, we emphasize from the outset that the basic ideas presented are generally applicable to almost any PDE system. The paper is really an introduction to the Discrete Calculus method. The fact that the diffusion equation is simple and has analogs in many fields of application should make the paper, and hence this method, available to a broad audience.

Two different node-centered Discrete Calculus methods are derived in detail (Section 3). The paper then shows how these ideas can be applied to cell-based discretizations and how they differ from traditional finite volume and discrete Galerkin methods (Section 4). Section 5 then compares these four Discrete Calculus methods to some classic finite volume methods for the diffusion equation on a variety of test problems.