When solving a general PDE such as

$$ \frac {\partial ^2 E}{\partial t ^2} = \frac {\partial ^2 E}{\partial z ^2} - \frac {\partial E}{\partial z} $$

this equation can be solved by the method of lines. The second order time-derivative can also be transformed into a first-order differential equation, but now the spatial derivatives have to approximated. Some methods such as backward and central finite difference approximations exist (for varying orders of accuracy), but are there any other possibilities to better approximation these spatial derivatives?

Also, on a related noted, when solving the above equation on a domain of $t$ = [0,1000] and $z$ = [0,1000], the steps can be made such that using a 1000x1000 grid, $\Delta z \sim 1$ and $\Delta t \sim 1$. But if one wanted to consider an arbitrary precision such as $\Delta z \sim 10^{-14}$, this results in very large oscillations (depending on the initial conditions). In the above equation, is there any particular limit for how small $\Delta z$ can be made? If I was to instead change the above into a system of first order differential equations in $z$ and discretized $t$, would the stability condition be analogous to reversing the Courant condition (i.e. becomes $c \Delta t > z$)?

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    $\begingroup$ I don't understand the first part of your question. You seem to ask what one would do with the spatial derivatives. There are of course a great many methods -- for example, finite differences, finite elements, finite volumes -- that can deal with this. What concretely is your question? $\endgroup$ – Wolfgang Bangerth Nov 14 '16 at 1:59
  • $\begingroup$ @WolfgangBangerth For the first part of the question, to treat the time derivatives, I was planning to use a finite difference method such as Runge-Kutta. But I was wondering if there were alternative options for evaluating the spatial derivatives specifically? A way to perhaps use Runge-Kutta on the time-derivatives and perhaps a more accurate method for the spatial derivatives besides central differences. $\endgroup$ – Mathews24 Nov 14 '16 at 3:33
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    $\begingroup$ Yes, as I said there are many. You could use 4th order finite differences for the spatial derivatives, or higher order finite elements, or spectral elements, or Fourier decompositions. So many options :-) $\endgroup$ – Wolfgang Bangerth Nov 14 '16 at 19:21
  • $\begingroup$ @WolfgangBangerth Thank you for the input. I have been using high order finite differences, but am uncertain of the total error accumulated in each step. At the moment, I am simply comparing different orders of accuracy. With regards to the discussion here, if I am marching $E$ in $z$ with an adaptive step, and both $N$ and $P$ in $t$ with uniform spacing, is there any particular suggested routine for approximating $\frac{\partial E}{\partial t}$? $\endgroup$ – Mathews24 Nov 14 '16 at 20:54
  • $\begingroup$ @WolfgangBangerth To my knowledge, spectral methods require nonuniform spacing in the variable considered, and thus would not necessarily work since I have an adaptive $z$-step; I am not familiar with any ways to implement both an adaptive $t-$ and $z$-step. Also, boundary/initial conditions appear to be very important. Would spectral methods or FEM be more accurate than higher order backward differences, especially if $\frac{\partial E}{\partial t}$ = $E$ = 0 for $(z,0)$ and $(0,t)$? $\endgroup$ – Mathews24 Nov 14 '16 at 20:56

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