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→Analysis: Correction to the local truncation error of the Euler Backwards method from O(h^2) to O(h). |
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{{Short description|Numerical method for ordinary differential equations}}
In [[numerical analysis]] and [[scientific computing]], the '''backward Euler method''' (or '''implicit Euler method''') is one of the most basic [[numerical methods for ordinary differential equations|numerical methods for the solution of ordinary differential equations]]. It is similar to the (standard) [[Euler method]], but differs in that it is an [[explicit and implicit methods|implicit method]]. The backward Euler method has error of order one in time.
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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
The [[Stiff_equation#Runge%E2%80%93Kutta_methods|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, 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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The
==See also==
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