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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Discrete Darboux polynomials and the preservation
of measure and integrals of ordinary differential
equations - Reinout Quispel (La Trobe University)
DTSTART;TZID=Europe/London:20190712T090000
DTEND;TZID=Europe/London:20190712T100000
UID:TALK127201AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/127201
DESCRIPTION:Preservation of phase space volume (or more genera
lly measure)\, first integrals (such as energy)\,
and second integrals have been important topics in
geometric numerical integration for more than a d
ecade\, and methods have been developed to preserv
e each of these properties separately. Preserving
two or more geometric properties simultaneously\,
however\, has often been difficult\, if not impos
sible. Then it was discovered that Kahan&rsquo\;s
&lsquo\;unconventional&rsquo\; method seems to pe
rform well in many cases [1]. Kahan himself\, howe
ver\, wrote: &ldquo\;I have used these unconventio
nal methods for 24 years without quite understandi
ng why they work so well as they do\, when they wo
rk.&rdquo\; The first approximation to such an un
derstanding in computational terms was: Kahan&rsq
uo\;s method works so well because

1. \;&
nbsp\; \; \; \; \; \; \; I
t is very successful at preserving multiple quanti
ties simultaneously\, eg modified energy and modi
fied measure.

2. \; \; \; \;&
nbsp\; \; \; \; It is linearly implici
t

3. \; \; \; \; \; \
; \; \; It is the restriction of a Runge-K
utta method

However\, point 1 above raises a
further obvious question: Why does Kahan&rsquo\;s
method preserve both certain (modified) first inte
grals and certain (modified) measures? In this t
alk we invoke Darboux polynomials to try and answe
r this question. The method of Darboux polynomial
s (DPs) for ODEs was introduced by Darboux to dete
ct rational integrals. Very recently we have advoc
ated the use of DPs for discrete systems [2\,3]. D
Ps provide a unified theory for the preservation o
f polynomial measures and second integrals\, as we
ll as rational first integrals. In this new perspe
ctive the answer we propose to the above question
is: Kahan&rsquo\;s method works so well because it
is good at preserving (modified) Darboux polynom
ials. If time permits we may discuss extensions t
o polarization methods. \;

[1] Petr
era et al\, Regular and Chaotic Dynamics 16 (2011)
\, 245&ndash\;289.

[2] Celledoni et al\, arxi
v:1902.04685.

[3] Celledoni et al\, arxiv:190
2.04715.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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