4.1.1 Euler-Cauchy Algorithm

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4.1.1 Euler-Cauchy Algorithm

Apply DNGF approximation

$\displaystyle \left. \frac{d\mbox{$\bf y$}}{dt} \right\vert _{t_{n}}$

$\textstyle =$

$\displaystyle \frac{\Delta \mbox{$\bf y$}_{n}}{\Delta t} + O[(\Delta t)]$

to the linear DE and find the Euler-Cauchy (EC) formula

$\displaystyle \frac{\Delta \mbox{$\bf y$}_{n}}{\Delta t}$

$\textstyle =$

$\displaystyle f_{n} + O[(\Delta t)]$

or

$\fbox{ \parbox{300pt}{ \begin{displaymath} \mbox{$\bf y$}_{n+1}=\mbox{$\bf y$}_{n}+\mbox{$\bf f$}_{n} \Delta t + O[(\Delta t)^{2}] \end{displaymath}} }$

Algebraically and computationally simple, but useless:

- Only first order accuracy

- Unstable: small aberrations from the true solution tend to grow in the course of further steps. $\Longrightarrow$

Apply EC to the relaxation equation $\frac{dy(t)}{dt}=-\lambda y(t)$ :

$\displaystyle y_{n+1}$

$\textstyle =$

$\displaystyle (1-\lambda \, \Delta t) \, y_{n}$

EC applied to the equation $dy/dt=-\lambda y$ , with $\lambda=1$ and $y_{0}=1$ : unstable for $\lambda \Delta t > 2$

Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001