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the local truncation error of implicit euler is O(h) not O(h^2) |
Jitse Niesen (talk | contribs) local error is O(h^2), see talk |
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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. This means that the [[local truncation error]] (defined as the error made in one step) is <math> O(h^2) </math>, using the [[big O notation]]. 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]],
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