Linear multistep method: Difference between revisions

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The result is approximations for the value of <math> y(t) </math> at discrete times <math> t_i </math>:

: <math> y_i \approx y(t_i) \quad\text{where}\quad t_i = t_0 + i h, </math>

: <math> y_i = y(t_i) = y(t_0 + i h) </math>

: <math> f_i = f(t_i, y_i) </math>

where ''h'' is the time step (sometimes referred to as <math> \Delta t </math>).

AMultistep methods use information from more than one prior step to calculate the next value. In particular, a ''linear'' multistep method uses a linear combination of <math>y_i</math> and <math>f(t_i,y_i')</math> to calculate the value of ''y'' for the desired current step. Thus, a linear multistep method is a method of the form

Multistep method will use the previous ''s'' steps to calculate the next value. Consequently, the desired value at the current processing stage is <math> y_{n+s} </math>.

A linear multistep method is a method of the form

: <math> \begin{align}

& y_{n+s} + a_{s-1} y_{n+s-1} + a_{s-2} y_{n+s-2} + \cdots + a_0 y_n \\

& \qquad {} = h \bigl( b_s f(t_{n+s},y_{n+s}) + b_{s-1} f(t_{n+s-1},y_{n+s-1}) + \cdots + b_0 f(t_n,y_n) \bigr),

\end{align} </math>

where ''h'' denotes the step size and ''f'' the right-hand side of the differential equation. The coefficients <math> a_0, \ldots, a_{s-1} </math> and <math> b_0, \ldots, b_s </math> determine the method. The designer of the method chooses the coefficients; often, manybalancing coefficientsthe areneed zero.to Typically,get thea designergood choosesapproximation to the coefficientstrue sosolution theyagainst willthe exactlydesire interpolateto <math>y(t)</math>get whena itmethod that is aneasy ''n''thto orderapply. polynomialOften, many coefficients are zero to simplify the method.

If <math> b_s = 0 </math>, then the method is called "explicit", since the formula can directly compute <math> y_{n+s} </math>.

One can distinguish between [[explicit and implicit methods]]. If <math> b_s = 0 </math>, then the method is called "explicit", since the formula can directly compute <math> y_{n+s} </math>. If <math> b_s \ne 0 </math> then the method is called "implicit", since the value of <math>y_{n+s}</math> depends on the value of <math> f(t_{n+s}, y_{n+s}) </math>, and the equation must be solved for <math> y_{n+s} </math>. [[Iterative methods]] such as [[Newton's method]] are often used to solve the implicit formula.

Sometimes an explicit multistep method is used to "predict" the value of <math>y_{n+s}</math>. That value is then used in an implicit formula to "correct" the value. The result is a [[Predictor-correctorpredictor–corrector method]].

==Examples==