Numerical Convergence Analysis of the Frank–Kamenetskii Equation

Abstract

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.

Keywords

• frank-kamenetskii
• crank-nicolson
• high-order finite differences
• convergence analysis