Linear multistep method: Difference between revisions

Revision as of 08:32, 8 February 2019

Revision as of 09:04, 8 February 2019

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The Adams–Moulton methods with ''s'' = 0, 1, 2, 3, 4 are ({{harvnb|Hairer|Nørsett|Wanner|1993|loc=§III.1}}; {{harvnb|Quarteroni|Sacco|Saleri|2000}}):

:<math> y_n &= y_{n-1} + h f(t_n???,y_n???) , \qquad\text{(</math> This is the [[Backward Euler method|backward Euler method)}\\]]

:<math> y_{n+1} &= y_n + \frac{1}{2} h \left( f(t_{n+1},y_{n+1}) + f(t_n,y_n) \right) , \qquad\text{(</math> This is the trapezoidal[[Trapezoidal rule (differential equations)}\\|trapezoidal rule]]

: <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( f(t_{n+1},y_{n+1}) + f(t_n,y_n) \right) , \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{9}{24} 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) , \\

y_{n+4} &= y_{n+3} + h \left( \frac{251}{720} f(t_{n+4?},y_{n+4?}) + \frac{646}{720} f(t_{n+3?},y_{n+3?}) - \frac{264}{720} f(t_{n+2?},y_{n+2?}) + \frac{106}{720} f(t_{n+1?},y_{n+1?}) - \frac{19}{720} f(t_n?,y_n?) \right) .

\end{align} </math>