Elliot J. Carr

Elliot J. Carr, Nathan G. March

We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the semi-analytical method, relative to the number of eigenvalues, is investigated and the effect of varying the interface conditions on the solution behaviour is explored. Numerical experiments demonstrate that solutions can be computed using the new semi-analytical method that are more accurate and more efficient than the unified transform method of Sheils [Appl. Math. Model., 46:450-464, 2017]. Furthermore, unlike classical analytical solutions and the unified transform method, only the new semi-analytical method is able to correctly treat problems with both time-varying external boundary conditions and a large number of layers. The paper is concluded by replicating solutions to several important industrial, environmental and biological applications previously reported in the literature, demonstrating the wide applicability of the work.

Nathan G. March, Elliot J. Carr

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 $τ$ satisfies the classical constraint of $τ\leq h_{i}^2/(2D_{i})$ 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.

Elliot J. Carr

Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a \textit{finite} transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable. In total, three approaches are studied: (i) mean action time (ii) mean plus one standard deviation of action time and (iii) a new approach derived by approximating the large time asymptotic behaviour of the cumulative distribution function. The new approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance $δ$ and the $(k-1)$th and $k$th moments ($k \geq 1$) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. Firstly, while the first two approaches are useful at characterising the time scale of the transition, they do not provide accurate estimates for diffusion processes. Secondly, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits, using only the moments, with the accuracy increasing as the index $k$ is increased.