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I'm working my way through these blog posts about the wave equation. All has made sense up until now.

The wave equation is $$ \frac{\partial^2h}{\partial t^2} = c^2 \frac{\partial^2h}{\partial x^2} $$

which, in discrete form, can be written as

$$a_i = c^2 \frac{h_{i+1} - 2h_i + h_{i- 1}}{\Delta x^2}$$

However, the author says that this an oversimplification and results in instability. The reasoning is "we have to consider that every point in the array of heights depends on not only its previous positions and velocities, but the heights and velocities and accelerations of neighboring points."

He then goes on to introduce a system of linear equations, but I simply don't "get" why the above equation is a simplification. If I know the height at point $i-1$, $i$, and $i+1$, I can calculate the second derivative wrt $x$, which (when multiplied by $c$) gives me the acceleration at that point, according to the equation above.

Please can anyone offer an explanation why this is an oversimplification?

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  • $\begingroup$ That's not how I interpret the statement This was the oversimplification in our previous attempt, and the source of the instability. Seems to me the sentence applies to the previous attempt (the previous lesson) and not the wave equation he's considering. $\endgroup$ – Kyle Kanos May 20 '15 at 20:59
  • $\begingroup$ The blog is clearly referring to the previous attempt to solve the discretized equation by means of an RK4 solver at every point. I don't know how to explicitly show the instability, but to say at least even though an RK4 method with an $O(\Delta^4)$ error estimate is used, the actual leading error will be $O(\Delta x^2)$ due to the spatial discretization. If one wants to economically improve the accuracy of the integration, one must use exactly the methods sketched in the blog. $\endgroup$ – Void May 20 '15 at 21:43
  • $\begingroup$ You may have more luck with answers on this site: scicomp.stackexchange.com $\endgroup$ – Ján Lalinský May 20 '15 at 22:01
  • $\begingroup$ @Void: Instabilities can be determined from von Neumann stability analysis. Also, the RK4 is the time integration, so that should be $\mathcal{O}(\Delta t^4)$, no? (you're missing a variable there) $\endgroup$ – Kyle Kanos May 20 '15 at 22:23

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