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Next: 4.2.4 Runge-Kutta Method for Up: 4.2 Initial Value Problems Previous: 4.2.2 Predictor-Corrector Method for


4.2.3 Nordsieck Formulation of the PC Method

Instead of threading a NGB polynomial through preceding points, expand $y(t)$ about $t_{n}$. $\Longrightarrow$ ``Taylor predictor''(e.g. of order 3):
$\displaystyle y_{n+1}^{P}$ $\textstyle =$ $\displaystyle y_{n}+\dot{y}_{n}\Delta t
+\stackrel{..}{y}_{n}\frac{(\Delta t)^{...
...\stackrel{...}{y}_{n}\frac{(\Delta t)^{3}}{3!} + O[(\Delta t)^{4}]
\hspace{1em}$  
$\displaystyle \dot{y}_{n+1}^{P} \Delta t$ $\textstyle =$ $\displaystyle \hspace{2.5em}\dot{y}_{n}\Delta t
+\stackrel{..}{y}_{n}(\Delta t)^{2}
+\stackrel{...}{y}_{n}\frac{(\Delta t)^{3}}{2!} + O[(\Delta t)^{4}]$  
$\displaystyle \ddot{y}_{n+1}^{P} \frac{(\Delta t)^{2}}{2!}$ $\textstyle =$ $\displaystyle \hspace{6em} \ddot{y}_{n} \frac{(\Delta t)^{2}}{2!}
+\stackrel{...}{y}_{n}\frac{(\Delta t)^{3}}{2!} + O[(\Delta t)^{4}]$  
$\displaystyle \stackrel{...}{y}_{n+1}^{P} \frac{(\Delta t)^{3}}{3!}$ $\textstyle =$ $\displaystyle \hspace{10.5em} \stackrel{...}{y}_{n}\frac{(\Delta t)^{3}}{3!}
+ O[(\Delta t)^{4}]$  



Defining the vector
$\displaystyle \mbox{$\bf z$}_{n} \equiv \left( \begin{array}{c}
y_{n} \\  \dot{...
...\frac{\textstyle (\Delta t)^{2}}
{\textstyle 2!} \\
\vdots \end{array} \right)$      

and the (Pascal triangle) matrix
$\displaystyle \mbox{${\bf A}$}$ $\textstyle \equiv$ $\displaystyle \left(
\begin{array}{ccccc}
1 & 1 & 1 & 1 & \dots \\
0 & 1 & 2 &...
... \dots \\
& \ddots & \ddots & 1 & \ddots \\
& & & & \ddots
\end{array}\right)$  

we have
$\displaystyle \fbox{$ \displaystyle
\mbox{$\bf z$}_{n+1}^{P}=\mbox{${\bf A}$}\cdot \mbox{$\bf z$}_{n}
$}$      

Now evaluate: $\Longrightarrow$

Evaluation step:

Insert $\mbox{$\bf z$}_{n+1}^{P}$ in the force law: $
b_{n+1}^{P} \equiv b[y_{n+1}^{P}, \dot{y}_{n+1}^{P}]
$

Corrector step:

Define the deviation $
\gamma \equiv [b_{n+1}^{P} - \ddot{y}_{n+1}^{P}] \frac{(\Delta t)^{2}}{2}
$ and write the corrector as

$\displaystyle \fbox{$ \displaystyle
\mbox{$\bf z$}_{n+1}=\mbox{$\bf z$}_{n+1}^{P}+\gamma \mbox{$\bf c$}
$}$      

with an optimized coefficient vector $\mbox{$\bf c$}$. The first few vectors are
$\displaystyle \mbox{$\bf c$}
=\left( \begin{array}{c} 1/6 \\  5/6 \\  1 \\  1/3...
...} 3/20 \\  251/360 \\  1 \\  11/18 \\  1/6 \\  1/60
\end{array} \right)\,,\dots$      



Advantages of Nordsieck PC:

- Self-starting
- Adjustable time steps

Stability: Again, between explicit (bad) and implicit (good).


next up previous
Next: 4.2.4 Runge-Kutta Method for Up: 4.2 Initial Value Problems Previous: 4.2.2 Predictor-Corrector Method for
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001