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4.2.2 Predictor-Corrector Method for 2nd order ODE

Predictor step:

In $d^{2}y/dt^{2}=b(t)$ replace the function $b(t)$ by a NGB polynomial and integrate twice.
$\displaystyle \dot{y}^{P}_{n+1}\Delta t-\dot{y}_{n}\Delta t$ $\textstyle =$ $\displaystyle (\Delta t)^{2}
\left[ b_{n} + \frac{1}{2} \nabla b_{n} + \frac{5}{12} \nabla^{2} b_{n} +
\frac{3}{8} \nabla^{3} b_{n} + \dots \right]$  
       
$\displaystyle y^{P}_{n+1}\!-\!y_{n}\!-\!\dot{y}_{n} \Delta t$ $\textstyle =$ $\displaystyle \frac{(\Delta t)^{2}}{2}
\left[ b_{n} + \frac{1}{3} \nabla b_{n}
+ \frac{1}{4} \nabla^{2} b_{n} + \frac{19}{90} \nabla^{3} b_{n}
+ \dots \right]$  

A specific predictor of order $k$ is found by using terms up to order $\nabla^{k-2}b_{n}$. Thus the predictor of third order reads
$\displaystyle \dot{y}^{P}_{n+1}\Delta t-\dot{y}_{n}\Delta t$ $\textstyle =$ $\displaystyle (\Delta t)^{2}
\left[ \frac{3}{2}b_{n} - \frac{1}{2} b_{n-1} \right] + O[(\Delta t)^{4}]$  
$\displaystyle y^{P}_{n+1}-y_{n}-\dot{y}_{n}\Delta t$ $\textstyle =$ $\displaystyle \frac{(\Delta t)^{2}}{2}
\left[ \frac{4}{3} b_{n} - \frac{1}{3} b_{n-1} \right]
+ O[(\Delta t)^{4}]$  



For a compact notation we define the vector
$\displaystyle \mbox{$\bf b$}_{k}$ $\textstyle \equiv$ $\displaystyle \{ b_{n},b_{n-1},\dots b_{n-k+2} \}^{T}$  

and the coefficient vectors $\mbox{$\bf c$}_{k}$ and $\mbox{$\bf d$}_{k}$. Then

\fbox{ \begin{minipage}{450pt}
{\bf Predictor of order k for second order DE:}
\...
... \mbox{$\bf b$}_{k}+O[(\Delta t)^{k+1}]
\nonumber \end{eqnarray}\end{minipage}}

The first few vectors $\mbox{$\bf c$}_{k},\mbox{$\bf d$}_{k}$ are given by
$\displaystyle \mbox{$\bf c$}_{2}=1 \hspace{5.5em}$   $\displaystyle \;\; \mbox{$\bf d$}_{2}=1$  
       
$\displaystyle \mbox{$\bf c$}_{3}=\left( \begin{array}{r} 3/2 \\  -1/2 \end{array} \right)
\hspace{1.5em}$   $\displaystyle \;\;
\mbox{$\bf d$}_{3}=\left( \begin{array}{r} 4/3 \\  -1/3 \end{array} \right)$  
$\displaystyle \mbox{$\bf c$}_{4}=\left( \begin{array}{r} 23/12 \\  -16/12 \\  5/12
\end{array} \right) \;$   $\displaystyle \;\;
\mbox{$\bf d$}_{4}=\left( \begin{array}{r} 19/12 \\  -10/12 \\  3/12
\end{array} \right)$ (4.1)
       
$\displaystyle \mbox{$\bf c$}_{5}=\left( \begin{array}{r} 55/24 \\  -59/24 \\  37/24 \\  -9/24
\end{array}\right) \;$   $\displaystyle \;\;
\mbox{$\bf d$}_{5}=\left( \begin{array}{r} 323/180 \\  -264/180 \\  159/180 \\  -38/180
\end{array} \right)$  



Evaluation step:

Insert the preliminary result $y_{n+1}^{P},\dot{y}_{n+1}^{P}$ in the physical law for $b[y,\dot{y}]$:
$\displaystyle b_{n+1}^{P}$ $\textstyle \equiv$ $\displaystyle b\left[ y_{n+1}^{P},\dot{y}_{n+1}^{P}\right]$  

Corrector step:

Insert $b_{n+1}^{P}$ in a NGB formula centered on $t_{n+1}$ and re-integrate twice:

$\displaystyle \dot{y}_{n+1}\Delta t - \dot{y}_{n}\Delta t$ $\textstyle =$ $\displaystyle (\Delta t)^{2}
\left[ b_{n+1}^{P} - \frac{1}{2} \nabla b_{n+1}
- \frac{1}{12} \nabla^{2} b_{n+1}
- \frac{1}{24} \nabla^{3} b_{n+1}
- \dots \right]$  
       
$\displaystyle y_{n+1} - y_{n} - \dot{y}_{n}\Delta t$ $\textstyle =$ $\displaystyle \frac{(\Delta t)^{2}}{2}
\left[ b_{n+1}^{P} - \frac{2}{3} \nabla ...
...rac{1}{12} \nabla^{2} b_{n+1} - \frac{7}{180}\nabla^{3} b_{n+1}
- \dots \right]$  



Defining the vector
$\displaystyle \mbox{$\bf b$}_{k}^{P}$ $\textstyle \equiv$ $\displaystyle \{ b_{n+1}^{P},b_{n},\dots b_{n-k+3} \}^{T}$  

and coefficient vectors $\mbox{$\bf e$}_{k},\mbox{$\bf f$}_{k}$, we write

\fbox{ \begin{minipage}{450pt}
{\bf Corrector of order k for second-order DE:}
\...
...ox{$\bf b$}_{k}^{P}+O[(\Delta t)^{k+1}]
\nonumber \end{eqnarray}\end{minipage}}

The first few coefficient vectors are
$\displaystyle \mbox{$\bf e$}_{2}=1 \hspace{5em}$   $\displaystyle \;\; \mbox{$\bf f$}_{2}=1$  
       
$\displaystyle \mbox{$\bf e$}_{3}=\left( \begin{array}{r} 1/2 \\  1/2 \end{array}\right)
\hspace{1.8em}$   $\displaystyle \;\;
\mbox{$\bf f$}_{3}=\left( \begin{array}{r} 1/3 \\  2/3 \end{array}\right)$  
$\displaystyle \mbox{$\bf e$}_{4}=\left( \begin{array}{r} 5/12 \\  8/12 \\  -1/12
\end{array}\right) \;$   $\displaystyle \;\;
\mbox{$\bf f$}_{4}=\left( \begin{array}{r} 3/12 \\  10/12 \\  -1/12
\end{array}\right)$ (4.2)
       
$\displaystyle \mbox{$\bf e$}_{5}=\left( \begin{array}{r} 9/24 \\  19/24 \\  -5/24 \\  1/24
\end{array}\right) \;$   $\displaystyle \;\;
\mbox{$\bf f$}_{5}=\left( \begin{array}{r} 38/180 \\  171/180 \\  -36/180 \\  7/180
\end{array}\right)$  




next up previous
Next: 4.2.3 Nordsieck Formulation of Up: 4.2 Initial Value Problems Previous: 4.2.1 Verlet Method
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001