Numerical Convergence Analysis of the Frank-Kamenetskii Equation.

This work investigates the convergence dynamics of a numerical scheme employed for the approximation and solution of the Frank-Kamenetskii partial differential equation. A framework for computing the critical Frank-Kamenetskii parameter to arbitrary accuracy is presented and used in the subsequent numerical simulations. The numerical method employed is a Crank-Nicolson type implicit scheme coupled with a fourth order spatial discretisation as well as a Newton-Raphson update step which allows for the nonlinear source term to be treated implicitly. This numerical implementation allows for the analysis of the convergence of the transient solution toward the steady-state solution. The choice of termination criteria, numerically dictating this convergence, is interrogated and it is found that the traditional choice for termination is insufficient in the case of the Frank-Kamenetskii partial differential equation which exhibits slow transience as the solution approaches the steady-state. Four measures of convergence are proposed, compared and discussed herein.

Nonstandard Finite Difference Method Applied to a Linear Pharmacokinetics Model.

We extend the nonstandard finite difference method of solution to the study of pharmacokinetic-pharmacodynamic models. Pharmacokinetic (PK) models are commonly used to predict drug concentrations that drive controlled intravenous (I.V.) transfers (or infusion and oral transfers) while pharmacokinetic and pharmacodynamic (PD) interaction models are used to provide predictions of drug concentrations affecting the response of these clinical drugs. We structure a nonstandard finite difference (NSFD) scheme for the relevant system of equations which models this pharamcokinetic process. We compare the results obtained to standard methods. The scheme is dynamically consistent and reliable in replicating complex dynamic properties of the relevant continuous models for varying step sizes. This study provides assistance in understanding the long-term behavior of the drug in the system, and validation of the efficiency of the nonstandard finite difference scheme as the method of choice.