On conservative, stable boundary and coupling conditions for diffusion equations I -- The conservation property for explicit schemes

TL;DR

This paper develops explicit numerical schemes for diffusion equations that preserve conservation properties at boundaries and interfaces, addressing a key oversight in existing methods, with a focus on finite difference and finite volume approaches.

Contribution

It introduces and analyzes explicit schemes that ensure conservation in boundary and coupling conditions, filling a gap in the numerical treatment of diffusion equations.

Findings

Discrete coupling conditions preserve conservation in nodal schemes.

Finite volume schemes effectively uphold conservation at interfaces.

Theoretical results are validated through numerical test cases.

Abstract

This paper introduces improved numerical techniques for addressing numerical boundary and interface coupling conditions in the context of diffusion equations in cellular biophysics or heat conduction problems in fluid-structure interactions. Our primary focus is on two critical numerical aspects related to coupling conditions: the preservation of the conservation property and ensuring stability. Notably, a key oversight in some existing literature on coupling methods is the neglect of upholding the conservation property within the overall scheme. This oversight forms the central theme of the initial part of our research. As a first step, we limited ourselves to explicit schemes on uniform grids. Implicit schemes and the consideration of varying mesh sizes at the interface will be reserved for a subsequent paper \cite{CMW3}. Another paper \cite{CMW2} will address the issue of stability. We examine these schemes from the perspective of finite differences, including finite elements, following the application of a nodal quadrature rule. Additionally, we explore a finite volume-based scheme involving cells and flux considerations. Our analysis reveals that discrete boundary and flux coupling conditions uphold the conservation property in distinct ways in nodal-based and cell-based schemes. The coupling conditions under investigation encompass well-known approaches such as Dirichlet-Neumann coupling, heat flux coupling, and specific channel and pumping flux conditions drawn from the field of biophysics. The theoretical findings pertaining to the conservation property are corroborated through computations across a range of test cases.