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General linear methods

Class of numerical method to solve differential equations

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Source updated March 30, 2026

General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to ordinary differential equations. They include 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).

01Some definitions

Numerical methods for first-order ordinary differential equations approximate solutions to initial value problems of the form

$y'=f(t,y),\quad y(t_{0})=y_{0}.$

The result is approximations for the value of $y(t)$ at discrete times $t_{i}$ :

$y_{i}\approx y(t_{i})\quad {\text{where}}\quad t_{i}=t_{0}+ih,$

where h is the time step (sometimes referred to as $\Delta t$ ).

02A description of the method

We follow Butcher (2006), pp. 189-190 for our description, although we note that this method can be found elsewhere.

General linear methods make use of two integers: $r$ , the number of time points in history, and $s$ , the number of collocation points. In the case of $r=1$ , these methods reduce to classical Runge-Kutta methods, and in the case of $s=1$ , these methods reduce to linear multistep methods.

Stage values $Y_{i}$ and stage derivatives $F_{i},\ i=1,2,\dots s$ are computed from approximations $y_{i}^{[n-1]},\ i=1,\dots ,r$ at time step $n$ :

$y^{[n-1]}=\left[{\begin{matrix}y_{1}^{[n-1]}\\y_{2}^{[n-1]}\\\vdots \\y_{r}^{[n-1]}\\\end{matrix}}\right],\quad y^{[n]}=\left[{\begin{matrix}y_{1}^{[n]}\\y_{2}^{[n]}\\\vdots \\y_{r}^{[n]}\\\end{matrix}}\right],\quad Y=\left[{\begin{matrix}Y_{1}\\Y_{2}\\\vdots \\Y_{s}\end{matrix}}\right],\quad F=\left[{\begin{matrix}F_{1}\\F_{2}\\\vdots \\F_{s}\end{matrix}}\right]=\left[{\begin{matrix}f(Y_{1})\\f(Y_{2})\\\vdots \\f(Y_{s})\end{matrix}}\right].$

The stage values are defined by two matrices $A=[a_{ij}]$ and $U=[u_{ij}]$ :

$Y_{i}=\sum _{j=1}^{s}a_{ij}hF_{j}+\sum _{j=1}^{r}u_{ij}y_{j}^{[n-1]},\qquad i=1,2,\dots ,s,$

and the update to time $t^{n}$ is defined by two matrices $B=[b_{ij}]$ and $V=[v_{ij}]$ :

$y_{i}^{[n]}=\sum _{j=1}^{s}b_{ij}hF_{j}+\sum _{j=1}^{r}v_{ij}y_{j}^{[n-1]},\qquad i=1,2,\dots ,r.$

Given the four matrices $A,U,B$ and $V$ , one can compactly write the analogue of a Butcher tableau as

$\left[{\begin{matrix}Y\\y^{[n]}\end{matrix}}\right]=\left[{\begin{matrix}A\otimes I&U\otimes I\\B\otimes I&V\otimes I\end{matrix}}\right]\left[{\begin{matrix}hF\\y^{[n-1]}\end{matrix}}\right],$

where $\otimes$ stands for the Kronecker product.

03Examples

We present an example described in (Butcher, 1996). This method consists of a single "predicted" step and "corrected" step, which 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:

$y_{n-1/2}^{*}=y_{n-2}+h\left({\frac {9}{8}}f(y_{n-1})+{\frac {3}{8}}f(y_{n-2})\right).$

An initial "predictor" $y_{n}^{*}$ uses the stage value $y_{n-1/2}^{*}$ together with two pieces of time history:

$y_{n}^{*}={\frac {28}{5}}y_{n-1}-{\frac {23}{5}}y_{n-2}+h\left({\frac {32}{15}}f(y_{n-1/2}^{*})-4f(y_{n-1})-{\frac {26}{15}}f(y_{n-2})\right),$

and the final update is given by

$y_{n}={\frac {32}{31}}y_{n-1}-{\frac {1}{31}}y_{n-2}+h\left({\frac {5}{31}}f(y_{n}^{*})+{\frac {64}{93}}f(y_{n-1/2}^{*})+{\frac {4}{31}}f(y_{n-1})-{\frac {1}{93}}f(y_{n-2})\right).$

The concise table representation for this method is given by

$\left[{\begin{array}{ccc|cccc}0&0&0&0&1&{\frac {9}{8}}&{\frac {3}{8}}\\{\frac {32}{15}}&0&0&{\frac {28}{5}}&-{\frac {23}{5}}&-4&-{\frac {26}{15}}\\{\frac {64}{93}}&{\frac {5}{31}}&0&{\frac {32}{31}}&-{\frac {1}{31}}&{\frac {4}{31}}&-{\frac {1}{93}}\\\hline {\frac {64}{93}}&{\frac {5}{31}}&0&{\frac {32}{31}}&-{\frac {1}{31}}&{\frac {4}{31}}&-{\frac {1}{93}}\\0&0&0&1&0&0&0\\0&0&1&0&0&0&0\\0&0&0&0&0&1&0\\\end{array}}\right].$

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Sources and credits

This article is adapted from the Wikipedia article “General linear methods”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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General linear methods

01Some definitions

02A description of the method

03Examples

Sources and credits

Related topics

Runge-Kutta methods

Linear multistep method

Numerical methods for ordinary differential equations