Finite volume schemes for multilayer diffusion

TL;DR

This paper introduces a stable and accurate finite volume scheme for multilayer diffusion problems, providing stability analysis for various time discretizations and demonstrating effectiveness through numerical examples.

Contribution

The work develops a finite volume method that maintains second order accuracy, addresses interface condition limitations, and provides stability analysis for multiple time discretizations.

Findings

Backward Euler and Crank-Nicolson schemes are unconditionally stable.

Forward Euler stability depends on more restrictive conditions than classical constraints.

Numerical examples confirm the theoretical stability and accuracy of the proposed method.

Abstract

This paper focusses on finite volume schemes for solving multilayer diffusion problems. We develop a finite volume method that addresses a deficiency of recently proposed finite volume/difference methods, which consider only a limited number of interface conditions and do not carry out stability or convergence analysis. Our method also retains second order accuracy in space while preserving the tridiagonal matrix structure of the classical single-layer discretisation. Stability and convergence analysis of the new finite volume method is presented for each of the three classical time discretisation methods: forward Euler, backward Euler and Crank-Nicolson. We prove that both the backward Euler and Crank-Nicolson schemes are always unconditionally stable. The key contribution of the work is the presentation of a set of sufficient stability conditions for the forward Euler scheme. Here, we find that to ensure stability of the forward Euler scheme it is not sufficient that the time step $\tau$ satisfies the classical constraint of $\tau\leq h_{i}^2/(2D_{i})$ in each layer (where $D_{i}$ is the diffusivity and $h_{i}$ is the grid spacing in the $i$th layer) as more restrictive conditions can arise due to the interface conditions. The paper concludes with some numerical examples that demonstrate application of the new finite volume method, with the results presented in excellent agreement with the theoretical analysis.