Backward Euler method: Difference between revisions

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== Derivation ==

Integrating the differential equation <math> \frac{\mathrm{d} y}{\mathrm{d} t} = f(t,y) </math> from <math> t_kt_n </math> to <math> t_{kn+1} = t_kt_n + h </math> yields

: <math> y(t_{kn+1}) - y(t_kt_n) = \int_{t_kt_n}^{t_{kn+1}} f(t, y(t)) \,\mathrm{d}t. </math>

Now approximate the integral on the right by the right-hand [[rectangle method]] (with one rectangle):

: <math> y(t_{kn+1}) - y(t_kt_n) \approx h f(t_{kn+1}, y(t_{kn+1})). </math>

Finally, use that <math> y_ky_n </math> is supposed to approximate <math> y(t_kt_n) </math> and the formula for the backward Euler method follows.<ref>{{harvnb|Butcher|2003|p=57}}</ref>

The same reasoning leads to the (standard) Euler method if the left-hand rectangle rule is used instead of the right-hand one.