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### Dispersjonsanalyse av differansemetode

Skrevet: 18/03-2019 00:39
Jeg jobber med Wendroff's metode:

$$U_{k}^{n+1}=U_{k-1}^{n}-\frac{1-r}{1+r}(U_{k-1}^{n+1}-U_k^n)$$
I approksimasjonen

$$u_k_n=u^ e^{i(\omega n \Delta t+\beta k \Delta x)}$$

med $$\omega=\alpha+ib$$, trenger jeg å vise at hvorvidt $$\alpha$$ er en triviell funksjon av $$\beta$$, altså hvorvidt

\frac{\partial^2}{\partial \beta^2} (\frac{\alpha}{\beta})=0

If the above second derivative is 0, then $\alpha$ is trivial and the difference method is non-dispersive. I've inserted (2) into (1) and found

e^{i\omega \Delta t}=e^{-i\beta\Delta x}-\frac{1-r}{1+r}(e^{i(\omega\Delta t-\beta\Delta x)}-1)

I'm not sure how to simplify this expression. I've seen one proof, for Lax-Wendroff method, that takes $e^{i\alpha\Delta t}$ and somehow constructs tan$(\alpha\Delta t)$ out of it. Others use Euler's formula to expand and then simplify. Any suggestions on how to proceed?
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