Further Comparisons of Finite Difference Schemes for Computational Modelling of Biosensors

Abstract

Simulations are presented for a reaction-diffusion system within a thin layer containing an enzyme, fed with a substrate from the surrounding electrolyte. The chemical term is of the nonlinear Michaelis-Menten type and requires a technique such as Newton iteration for solution. It is shown that approximating the nonlinear chemical term in these systems by a linearised form reduces both the accuracy and, in the case of second-order methods such as Crank-Nicolson, reduces the global error order from O(δT2) to O(δT). The first-order methods plain backwards implicit with and without linearisation, and Crank-Nicolson with linearisation are all of O(δT) and very similar in performance, requiring, for a given accuracy target, an order of magnitude more CPU time than the efficient methods backward implicit with extrapolation and Crank-Nicolson, both with Newton iteration to handle the nonlinearity. Steady state computations agree with expectations, tending to the known solutions for limiting cases. The Crank-Nicolson method shows some concentration oscillations close to the outer layer boundary but this does not propagate to the inner boundary at the electrode. The backward implicit methods do not result in such oscillations and if concentration profiles are of interest, may be preferred.