Linear multistep method: Difference between revisions

Revision as of 14:06, 19 March 2012

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==First and second Dahlquist barriers==

These two results were proved by [[Germund Dahlquist]] and represent an important bound for the order of convergence and for the [[Stiff equation#A-stability|A-stability]] of a linear multistep method.

===First Dahlquist barrier===

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===Second Dahlquist barrier===

There are no explicit [[Stiff equation#A-stability|A-stable]] and linear multistep methods. The implicit ones have order of convergence at most 2. The [[trapezoidal rule (differential equations)|trapezoidal rule]] has the smallest error constant amongst the A-stable linear multistep methods of order 2. These results were proven by {{harvharvtxt|HairerDahlquist|Wanner|1996|loc=Thm V.1.41963}}.

== See also ==

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* {{citation | first1 = Francis | last1 = Bashforth | year = 1883 | title = An Attempt to test the Theories of Capillary Action by comparing the theoretical and measured forms of drops of fluid. With an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, by J. C. Adams | ___location = Cambridge }}.

* {{Citation | last1=Butcher | first1=John C. | author1-link = John C. Butcher | title=Numerical Methods for Ordinary Differential Equations | publisher=John Wiley | isbn=978-0-471-96758-3 | year=2003}}.

* {{Citation | last1=Dahlquist | first1=Germund | author1-link=Germund Dahlquist | title=A special stability problem for linear multistep methods | doi=10.1007/BF01963532 | year=1963 | journal=BIT | issn=0006-3835 | volume=3 | pages=27–43}}.

* {{citation | first1 = Herman H. | last1 = Goldstine | author1-link = Herman Goldstine | year = 1977 | title = A History of Numerical Analysis from the 16th through the 19th Century | publisher = Springer-Verlag | ___location = New York | isbn = 978-0-387-90277-7 }}.

* {{citation | first1 = Ernst | last1 = Hairer | first2 = Syvert Paul | last2 = Nørsett | first3 = Gerhard | last3 = Wanner | year = 1993 | title = Solving ordinary differential equations I: Nonstiff problems | edition = 2nd | publisher = Springer Verlag | ___location = Berlin | isbn = 978-3-540-56670-0 }}.