Linear multistep method: Difference between revisions

Revision as of 04:09, 13 October 2010

Revision as of 04:17, 13 October 2010

Line 23:

where ''h'' denotes the step size and ''f'' the right-hand side of the differential equation. The coefficents <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, many coefficients are zero. Typically, the designer chooses the coefficients so they will exactly interpolate <math>y(t)</math> when it is an ''n''th order polynomial.

If the value of <math>b_s</math> is nonzero, then the value of <math>y_{n+s}</math> depends on the value of <math> f(t_{n+s}, y_{n+s}) </math>. Consequently, the method is [[explicit]] if <math> b_s = 0 </math>. In that case, the formula can directly compute <math> y_{n+s} </math>. If <math> b_s \ne 0 </math> then the method is [[implicit]] and the equation for <math> y_{n+s} </math> must be solved. [[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-corrector method]].

Line 98:

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

== Analysis ==

Line 135:

==First and second Dahlquist barriers==

These two results were proved by [[Germund Dahlquist]] and represent an important bound for the order of convergence and for the [[Stiff_equationStiff equation#A-stability|A-stability]] of a linear multistep method.

===First Dahlquist barrier===

Line 143:

===Second Dahlquist barrier===

There are no explicit [[Stiff_equationStiff equation#A-stability|A-stable]] and linear multistep methods. The implicit ones have order of convergence at most 2 {{harv|Hairer|Wanner|1996|loc=Thm V.1.4}}.

== See also ==