12 pm Thursday, 26 April, 2012
|For classical numerical methods for ordinary differential equations,|
there are inherent difficulties in constructing efficient
and reliable variable order and variable stepsize algorithms. Denote
by h the current stepsize and p the order of the current step.
In the case of Runge-Kutta methods, for example, embedding
is not able to provide asymtotically correct error estimates at
a moderate cost. On the other hand, for linear multistep methods,
the difficulties lie in the need to interpolate and reorganize the data
when a variation of h or p is called for.
Many of these difficulties can be overcome in the case of multistep-multivalue,
or general linear, methods. The underlying idea is the use of a Lagrange
multiplier formulation to provide a rational process for choosing among
competing p and selecting the optimal h value. A second underlying
idea is the use of the so-called `scale and modify technique to guarantee
smooth transitions between one step and the next, even though
h, and possibly also p, has been adjusted.