Linear multistep method: Difference between revisions

Revision as of 19:08, 26 September 2017

Revision as of 19:18, 26 September 2017

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: <math> \begin{align}

y_n &= y_{n-1} + h f(t_n,y_n) , \qquad\text{(This is the backward Euler method)}\\

y_{n+1} &= y_n + \frac{1}{2} h \left(Big[ f(t_{n+1},y_{n+1}) + f(t_n,y_n) \right)Big] , \qquad\text{(This is the trapezoidal rule)}\\

y_{n+2} &= y_{n+1} + h \left( \frac{5}{12} f(t_{n+2},y_{n+2}) + \frac{2}{3} f(t_{n+1},y_{n+1}) - \frac{1}{12} f(t_n,y_n) \right) , \\

y_{n+3} &= y_{n+2} + h \left( \frac{3}{8} f(t_{n+3},y_{n+3}) + \frac{19}{24} f(t_{n+2},y_{n+2}) - \frac{5}{24} f(t_{n+1},y_{n+1}) + \frac{1}{24} f(t_n,y_n) \right) , \\

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The derivation of the Adams–Moulton methods is similar to that of the Adams–Bashforth method; however, the interpolating polynomial uses not only the points <math>t_{n-1},\dots, t_{n-s} </math>, as above, but also <math> t_n </math>. The coefficients are given by

:<math>

:<math> b_{s-j} = \frac{(-1)^j}{j!(s-j)!} \int_0^1 \prod_{i=0 \atop i\ne j}^{s} (u+i-1) \, \mathrm du, \qquad \text{for } \ j=0,\,\ldots\,,s\,. </math>

</math>

The Adams–Moulton methods are solely due to [[John Couch Adams]], like the Adams–Bashforth methods. The name of [[Forest Ray Moulton]] became associated with these methods because he realized that they could be used in tandem with the Adams–Bashforth methods as a [[Predictor-corrector method|predictor-corrector]] pair {{harv|Moulton|1926}}; {{harvtxt|Milne|1926}} had the same idea. Adams used [[Newton's method]] to solve the implicit equation {{harv|Hairer|Nørsett|Wanner|1993|loc=§III.1}}.