Collocation method: Difference between revisions

Revision as of 10:17, 12 March 2012

Revision as of 14:51, 23 March 2012

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All these collocation methods are in fact implicit [[Runge–Kutta methods]]. However, not all implicit Runge–Kutta methods are collocation methods.

<ref>{{harvnb|Ascher|Petzold|1998}}; {{harvnb|Iserles|1996|pp=43–44}}</ref>

<ref>{{cite book |title= Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations |last= Ascher |first=Uri M. |authorlink= |coauthors=Linda R. Petzold |year=1998 |publisher=SIAM |___location=Philadelphia, USA |isbn=0-89871-412-5}}</ref>

=== Example ===

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and approximating the integral on the right-hand side by the [[trapezoidal rule]] for integrals.

== ReferencesNotes ==

{{Reflist}}

*Ernst Hairer, Syvert Nørsett and Gerhard Wanner, ''Solving ordinary differential equations I: Nonstiff problems,'' second edition, Springer Verlag, Berlin, 1993. ISBN 3-540-56670-8.

== References ==

*Arieh Iserles, ''A First Course in the Numerical Analysis of Differential Equations,'' Cambridge University Press, 1996. ISBN 0-521-55376-8 (hardback), ISBN 0-521-55655-4 (paperback).

<ref>* {{citeCitation book| last1=Ascher |title first1=Uri M. | last2=Petzold | first2=Linda R. | title=Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations |last= Ascher |firstpublisher=Uri[[Society M.for |authorlink=Industrial |coauthors=Lindaand R.Applied PetzoldMathematics]] |year=1998 |publisher=SIAM |___location=Philadelphia, USA| |isbn=978-0-89871-412-58 | year=1998}}</ref>.

*Ernst {{Citation | last1=Hairer, Syvert| first1=Ernst | last2=Nørsett and| Gerhardfirst2=Syvert Paul | last3=Wanner, ''| first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems,'' second edition,| publisher=[[Springer -Verlag,]] | ___location=Berlin, 1993.New ISBNYork | isbn=978-3-540-56670-80 | year=1993}}.

* {{Citation | last1=Iserles | first1=Arieh | author1-link=Arieh Iserles, ''| title=A First Course in the Numerical Analysis of Differential Equations,'' | publisher=[[Cambridge University Press,]] 1996.| ISBN 0isbn=978-521-55376-8 (hardback), ISBN 0-521-55655-42 (paperback)| year=1996}}.

{{Numerical PDE}}