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

TL;DR

The paper addresses the need for conservative, stable discretizations of coupling conditions in bi-domain diffusion problems under explicit time stepping. It develops and compares nodal FTCS and finite-volume schemes, deriving a discrete conservation property that ensures total mass is preserved when fluxes at boundaries and interfaces are discretized consistently. Through analytical arguments and numerical tests across Dirichlet-Neumann, heat-flux, channel pumping, and membrane pumping couplings, it demonstrates that conservative implementations (notably central boundary treatment in nodal schemes and flux-based FV schemes) maintain mass with high accuracy, while nonconservative approaches (as in Giles' scheme) can introduce mass loss. The findings have practical impact on simulations in biophysics and heat transfer, guiding the design of explicit schemes that respect fundamental conservation laws. The authors also indicate directions for future work on implicit schemes and stability analyses.

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.

Figures (8)

• Figure 1: Bi-domain cubic volume distribution of ER and cytosolic domains, modification of a figure from l9.
• Figure 2: Grid points and discrete values for bi-domain equations
• Figure 3: Cells, nodes, and cell or nodal values for the finite volume scheme.
• Figure 4: Bi-domain computations with six couplings: The interval [0,1] is divided into 100 sub-intervals of length $\Delta x=0.01$. There were $3000$ time steps of length $\Delta t=4\cdot 10^{-5}$, giving a final time $T=0.12$. The figure on the right is a zoom into the coupling region.
• Figure 5: Bi-domain computations with negative heat fluxes: The interval [0,1] is divided into 100,000 sub-intervals of length $\Delta x=0.00001$. There were $10^6$ time steps of length $\Delta t=4\cdot 10^{-11}$, giving a final time $T=0.00004$. The figure on the left is a zoom into the coupling region for the cosine initial data. The figure on the right is a zoom into the upper part of the coupling region for the piecewise constant initial data.