Slow-burning instabilities of Dufort-Frankel finite differencing

Dr. David Galloway (The University of Sydney)
Wednesday May 24, 2pm
Title:Slow-burning instabilities of Dufort-Frankel finite differencing.
Abstract: Du Fort-Frankel is a tactic to stabilise Richardson's unstable 3-level leapfrog time-stepping scheme. By including the next time level in the right hand side evaluation, it is implicit, but it can be re-arranged 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 teaching 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. Attempts to fix the rabies problem by additional averaging are described. One method works for a limited parameter range but beyond that, instability can take a very long time to appear and its analysis displays interesting subtleties. This is joint work with David Ivers.

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