General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to differential equations. This large class of methods in numerical analysis encompass multistage Runge–Kutta methods that use intermediate collocation points, as well as linear multistep methods that save a finite time history of the solution. John C. Butcher originally coined this term for these methods, and has written a series of review papers a book chapter and a textbook on the topic. His collaborator, Zdzislaw Jackiewicz also has an extensive textbook on the topic. The original class of methods were originally proposed by Butcher(1965), Gear (1965) and Gragg and Stetter (1964).
Contents
Some definitions
Numerical methods for first-order ordinary differential equations approximate solutions to initial value problems of the form
The result is approximations for the value of
where h is the time step (sometimes referred to as
A description of the method
We follow Butcher (2006), pps 189–190 for our description, although we note that this method can be found elsewhere.
General linear methods make use of two integers,
Stage values
The stage values are defined by two matrices,
and the update to time
Given the four matrices,
where
Examples
We present an example described in (Butcher, 1996). This method consists of a single 'predicted' step, and 'corrected' step, that uses extra information about the time history, as well as a single intermediate stage value.
An intermediate stage value is defined as something that looks like it came from a linear multistep method:
An initial 'predictor'
and the final update is given by:
The concise table representation for this method is given by: