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SLOW-BURNING INSTABILITIES OF DUFORT–FRANKEL FINITE DIFFERENCING

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  30 April 2021

DAVID GALLOWAY Open the ORCID record for DAVID GALLOWAY [Opens in a new window]  and
DAVID IVERS Open the ORCID record for DAVID IVERS [Opens in a new window]
DAVID GALLOWAY
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW2006, Australia; david.ivers@sydney.edu.au.
DAVID IVERS*
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW2006, Australia; david.ivers@sydney.edu.au.
*
Email: david.galloway@sydney.edu.au
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Abstract

DuFort–Frankel averaging is a tactic to stabilize Richardson’s unstable three-level leapfrog timestepping scheme. By including the next time level in the right-hand-side evaluation, it is implicit, but it can be rearranged to give an explicit updating formula, thus apparently giving the best of both worlds. Textbooks prove unconditional stability for the heat equation, and extensive use on a variety of advection–diffusion equations has produced many useful results. Nonetheless, for some problems the scheme can fail in an interesting and surprising way, leading to instability at very long times. An analysis for a simple problem involving a pair of evolution equations that describe the spread of a rabies epidemic gives insight into how this occurs. An even simpler modified diffusion equation suffers from the same instability. Finally, the rabies problem is revisited and a stable method is found for a restricted range of parameter values, although no prescriptive recipe is known which selects this particular choice.

Keywords

finite differencesnumerical instabilityDuFort–Frankelleapfrog schemesdiffusion

MSC classification

Primary: 65M12: Stability and convergence of numerical methods
Secondary: 65M06: Finite difference methods
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 1 , January 2021 , pp. 23 - 38
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Copyright
© Australian Mathematical Society 2021

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