Linear multistep method: Difference between revisions

Revision as of 20:56, 26 May 2021

Revision as of 22:39, 10 June 2021

Line 27:

Consider for an example the problem

: <math> y' = f(t,y)=y, \quad y(0) = 1. </math>

The exact solution is <math> y(t) = \mathrm{e}^t </math>.

=== One-step Euler ===

A simple numerical method is Euler's method:

: <math> y_{n+1} = y_n + hf(t_n, y_n). \, </math>

Euler's method can be viewed as an explicit multistep method for the degenerate case of one step.

Line 51:

y_4 &= y_3 + \tfrac32 hf(t_3, y_3) - \tfrac12 hf(t_2, y_2) = 3.7812 + \tfrac32\cdot\tfrac12\cdot3.7812 - \tfrac12\cdot\tfrac12\cdot2.375 = 6.0234.

\end{align} </math>

The exact solution at <math> t = t_4 = 2 </math> is <math> \mathrm{e}^2 = 7.3891\ldots </math>, so the two-step Adams–Bashforth method is more accurate than Euler's method. This is always the case if the step size is small enough.

== Families of multistep methods ==