General linear methods (GLMs) are a large class of
numerical methods
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
used to obtain
numerical solutions to
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s. They include multistage
Runge–Kutta methods that use intermediate
collocation points, as well as
linear multistep method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The proce ...
s that save a finite time history of the solution.
John C. Butcher
John Charles Butcher (born 31 March 1933) is a New Zealand mathematician who specialises in numerical methods for the solution of ordinary differential equations..
Butcher works on multistage methods for initial value problems, such as Runge- ...
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).
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
at discrete times
:
:
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,
, the number of time points in history and
, the number of collocation points. In the case of
, these methods reduce to classical
Runge–Kutta methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. The ...
,
and in the case of
, these methods reduce to
linear multistep method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The proce ...
s.
Stage values
and stage derivatives,
are computed from approximations,
, at time step
:
:
The stage values are defined by two matrices,
and