Backward Euler method: Difference between revisions

Revision as of 12:05, 24 March 2012

Revision as of 12:52, 24 March 2012

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[[File:Stability region for BDF1.svg|thumb|The pink region outside the disk shows the stability region of the backward Euler method.]]

The backward Euler method has order one. ItThis ismeans that the [[A-stability|A-stablelocal truncation error]] and(the evenerror [[L-stability|L-stable]],made makingin itone suitablestep) foris the<math> solutionO(h^2) of</math>, using the [[stiffbig equationO notation]]s. The error at a specific time <math> t </math> is <math> O(h) </math>.

The [[region of absolute stability]] for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure.<ref>{{harvnb|Butcher|2003|p=70}}</ref> This includes the whole left half of the complex plane, so the backward Euler method is [[A-stability|A-stable]], making it suitable for the solution of [[stiff equation]]s.<ref>{{harvnb|Butcher|2003|p=71}}</ref> In fact, the backward Euler method is even [[L-stability|L-stable]],

== Extensions and modifications ==