Numerical Convergence Analysis of the Frank–Kamenetskii Equation

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  • Mathew Woolway, University of the Witwatersrand, South Africa
  • Byron Jacobs, University of Witwatersrand, South Africa
  • Ebrahim Momoniat, University of the Witwatersrand, South Africa
  • Charis Harley, University of the Witwatersrand, WITS, South Africa
  • Dieter Britz
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.
Original languageEnglish
Article number84
Number of pages17
Publication statusPublished - 2020

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