general linear methods

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

: $y\text{'}\; =\; f(t,y),\; \backslash quad\; y(t\_0)\; =\; y\_0.$

The result is approximations for the value of $y(t)$ at discrete times $t\_i$:

: $y\_i\; \backslash approx\; y(t\_i)\; \backslash quad\backslash text\{where\}\backslash quad\; t\_i\; =\; t\_0\; +\; i\; h,$

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

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: $r${{snd}} the number of time points in history, and $s${{snd}} 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 $F\_i,\backslash \; i\; =\; 1,\; 2,\; \backslash dots\; s$ are computed from approximations $y\_i^\{[n-1]\},\backslash \; i\; =\; 1,\; \backslash dots,\; r$ 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} h F_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} h F_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}

h F \\

y^{[n-1]}

\end{matrix} \right],

where $\backslash otimes$ stands for the Kronecker product.

We present an example described in (Butcher, 1996).{{harvnb|Butcher|1996|p=107}}. 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( \frac9 8 f( y_{n-1} ) + \frac3 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} ) - 4 f( 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].

• Runge–Kutta methods
• Linear multistep methods
• Numerical methods for ordinary differential equations

{{Reflist}}

• {{cite journal|last=Butcher|first=John C.|title=A Modified Multistep Method for the Numerical Integration of Ordinary Differential Equations|journal=Journal of the ACM|date=January 1965|volume=12|issue=1|pages=124–135|doi=10.1145/321250.321261|s2cid=36463504|doi-access=free}}
• {{cite journal|last=Gear|first=C. W.|title=Hybrid Methods for Initial Value Problems in Ordinary Differential Equations|journal= Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis|year=1965|volume=2|issue=1|pages=69–86|doi=10.1137/0702006|hdl=2027/uiuo.ark:/13960/t4rj60q8s|bibcode=1965SJNA....2...69G|s2cid=122744897 |hdl-access=free}}
• {{cite journal|last=Gragg|first=William B.|author2=Hans J. Stetter|title=Generalized Multistep Predictor-Corrector Methods|journal=Journal of the ACM|date=April 1964|volume=11|issue=2|pages=188–209|doi=10.1145/321217.321223|s2cid=17118462|doi-access=free}}
• {{citation | first1 = Ernst | last1 = Hairer | first2 = Wanner | last2 = Wanner | year = 1973 | title = Multistep-multistage-multiderivative methods for ordinary differential equations | journal = Computing | volume = 11 | issue = 3 | pages = 287–303 |

doi = 10.1007/BF02252917| s2cid = 25549771 }}.

• [http://www.scholarpedia.org/article/General_linear_methods General Linear Methods]

{{Numerical integrators}}

Category:Numerical differential equations