General linear methods (GLMs) are a large class of
numerical methods 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 contras ...
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 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. Th ...
,
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
u_
U, or u, is the twenty-first letter and the fifth vowel letter of the Latin alphabet, used in the modern English alphabet and the alphabets of other western European languages and others worldwide. Its name in English is ''u'' (pronounced ...
/math>:
:
and the update to time
is defined by two matrices,
and
:
:
Given the four matrices,
and
, one can compactly write the analogue of a
Butcher tableau as,
:
where
stands for the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
.
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
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 ...
:
:
An initial 'predictor'
uses the stage value
together with two pieces of time history:
:
and the final update is given by:
:
The concise table representation for this method is given by:
:
See also
*
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. Th ...
*
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
*
Numerical methods for ordinary differential equations
Notes
References
*
*
*
* .
External links
General Linear Methods
{{Numerical integrators
Numerical differential equations