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1999 I 4A

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We have the system of equations for $p\left(t,x\right)$ with $c$ being a constant:

$\frac{\partial v}{\partial t}=\frac{\partial p}{\partial x}$ $\frac{\partial p}{\partial t}=c^{2}\frac{\partial v}{\partial x}$

And the finite difference scheme:

$v_{j}^{n+1}=v_{j}^{n}+\frac{\delta t}{2\delta x}\left[p_{j+1}^{n}-p_{j-1}^{n}\right]$ $p_{j}^{n+1}=p_{j}^{n}+\frac{c^{2}\delta t}{2\delta x}\left[v_{j+1}^{n}-v_{j-1}^{n}\right]$

If we substitute $v_{j}^{n}=\sum_{k}r^{n}\tilde{v}_{k}e^{2\pi ikj/N}$ , $p_{j}^{n}=\sum_{k}r^{n}\tilde{p}_{k}e^{2\pi ikj/N}$ :

$\tilde{v}_{k}r=\tilde{v}_{k}+\frac{\delta t}{2\delta x}\left[\tilde{p}_{k}e^{2\pi ik/N}-\tilde{p}_{k}e^{-2\pi ik/N}\right]=\tilde{v}_{k}+\frac{i\delta t}{\delta x}\tilde{p}_{k}\sin\theta_{k}$ $\tilde{p}_{k}r=\tilde{p}_{k}+\frac{c^{2}\delta t}{2\delta x}\left[\tilde{v}_{k}e^{2\pi ik/N}-\tilde{v}_{k}e^{-2\pi ik/N}\right]=\tilde{p}_{k}+\frac{ic^{2}\delta t}{\delta x}\tilde{v}_{k}\sin\theta_{k}$

With $θ k = 2π k / N$ . Combining the equations:

$\tilde{v}_{k}r=\tilde{v}_{k}+\frac{i\delta t}{\delta x}\left(\frac{ic^{2}\delta t}{\left(r-1\right)\delta x}\tilde{v}_{k}\sin\theta_{k}\right)\sin\theta_{k}$

This gives:

$0=r^{2}-2r+1+c^{2}\frac{\delta t^{2}}{\delta x^{2}}\sin^{2}\theta_{k}$

Solving for $r$ :

$r=1\pm i\left|c\frac{\delta t}{\delta x}\sin\theta_{k}\right|$

Or:

$r^{2}=1+c^{2}\frac{\delta t^{2}}{\delta x^{2}}\sin^{2}\theta_{k}$

This means that for finite $δ t$ , $δ x$ , $\left|r\right|$ is always $> 1$ for some $k$ , making the method unstable.

If we use the scheme:

$v_{j}^{n+1}=v_{j}^{n}+\frac{\delta t}{2\delta x}\left[p_{j+1}^{n+1/2}-p_{j-1}^{n+1/2}\right]$ $p_{j}^{n+1/2}=p_{j}^{n-1/2}+\frac{c^{2}\delta t}{2\delta x}\left[v_{j+1}^{n}-v_{j-1}^{n}\right]$

We get:

$\tilde{v}_{k}r=\tilde{v}_{k}+\frac{\delta t}{2\delta x}r^{1/2}\left[\tilde{p}_{k}e^{2\pi ik/N}-\tilde{p}_{k}e^{-2\pi ik/N}\right]=\tilde{v}_{k}+\frac{i\delta t}{\delta x}r^{1/2}\tilde{p}_{k}\sin\theta_{k}$ $\tilde{p}_{k}r=\tilde{p}_{k}+\frac{c^{2}\delta t}{2\delta x}r^{1/2}\left[\tilde{v}_{k}e^{2\pi ik/N}-\tilde{v}_{k}e^{-2\pi ik/N}\right]=\tilde{p}_{k}+\frac{ic^{2}\delta t}{\delta x}r^{1/2}\tilde{v}_{k}\sin\theta_{k}$

And then combining:

$\tilde{v}_{k}r=\tilde{v}_{k}+\frac{i\delta t}{\delta x}r\left(\frac{ic^{2}\delta t}{\left(r-1\right)\delta x}\tilde{v}_{k}\sin\theta_{k}\right)\sin\theta_{k}$

Or:

$0=r^{2}-2r+1+rc^{2}\frac{\delta t^{2}}{\delta x^{2}}\sin^{2}\theta_{k}$

Which gives:

$r=1-\frac{1}{2}c^{2}\frac{\delta t^{2}}{\delta x^{2}}\sin^{2}\theta_{k}\pm\left|c\frac{\delta t}{\delta x}\sin\theta_{k}\right|\sqrt{c^{2}\frac{\delta t^{2}}{\delta x^{2}}\sin^{2}\theta_{k}-2}$

At $θ k = π / 2$ :

$r=1-\frac{1}{2}c^{2}\frac{\delta t^{2}}{\delta x^{2}}\pm\left|c\frac{\delta t}{\delta x}\right|\sqrt{c^{2}\frac{\delta t^{2}}{\delta x^{2}}-2}$

So that, for $c^{2}\delta t^{2}\leq2\delta x^{2}$ :

$r^{2}=\left(1-\frac{1}{2}c^{2}\frac{\delta t^{2}}{\delta x^{2}}\right)^{2}+c^{2}\frac{\delta t^{2}}{\delta x^{2}}\left(2-c^{2}\frac{\delta t^{2}}{\delta x^{2}}\right)$

So that if:

$r^{2}=1+c^{2}\frac{\delta t^{2}}{\delta x^{2}}-\frac{3}{4}\left(c^{2}\frac{\delta t^{2}}{\delta x^{2}}\right)^{2}$

Our condition becomes:

$\frac{3}{4}\left(c^{2}\frac{\delta t^{2}}{\delta x^{2}}\right)^{2}\geq c^{2}\frac{\delta t^{2}}{\delta x^{2}}$

Or:

$c^{2}\frac{\delta t^{2}}{\delta x^{2}}\geq\frac{4}{3}$

For $c^{2}\delta t^{2}\geq2\delta x^{2}$ :

$r=\left(1-\frac{1}{2}c^{2}\frac{\delta t^{2}}{\delta x^{2}}\right)^{2}+\left|c\frac{\delta t}{\delta x}\right|^{2}\left(c^{2}\frac{\delta t^{2}}{\delta x^{2}}-2\right)$

Or: