General linear methods

Template:Distinguish2 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).

Some 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 $\Delta t$).

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, $r$, 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 $n$:

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

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 $V=[v_{ij}]$:

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

See also

• Runge-Kutta methods
• Linear multistep methods

References

• Butcher, John C. (1965). "A Modified Multistep Method for the Numerical Integration of Ordinary Differential Equations". Journal of the ACM (JACM). 12 (1): 124–135. doi:10.1145/321250.321261. {{cite journal}}: Unknown parameter |month= ignored (help)
• Gear, C.W. (1965). "Hybrid Methods for Initial Value Problems in Ordinary Differential Equations". Society for Industrial and Applied Mathematics. 2 (1): 69–86. doi:10.1137/0702006.
• Gragg, William B. (1964). "Generalized Multistep Predictor-Corrector Methods". Journal of the ACM (JACM). 11 (2): 188–209. doi:10.1145/321217.321223. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
• Hairer, Ernst,; Wanner, Wanner (1973), "Multistep-multistage-multiderivative methods for ordinary differential equations", Computing, Volume 11 (3): 287–303, doi:10.1007/BF02252917 {{citation}}: |volume= has extra text (help)CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link).

External links

• General Linear Methods