When solving an equation of the following form:$$ \begin{aligned}\frac {\partial A}{\partial t} &= EB - A \\\frac {\partial B}{\partial t} &= EA - B\\\frac {\partial E}{\partial t} &= u \\\frac {\partial u}{\partial t} &= u + \frac{1}{10^4}\frac {\partial E}{\partial z} - \frac{1}{10^8}\frac {\partial^2 E}{\partial z^2} - \frac {\partial^2 B}{\partial t^2} - \frac {\partial B}{\partial t}\end{aligned}$$I am able to produce the desired output that matches figures in this text when applying a Cash-Karp method and approximating the spatial derivatives by central differences. (Backward difference approximations of order 5 for the spatial derivatives appear to work, too). The domain I am considering in the above case is from $t = [0,1000] \times z = [0,1000]$ and my step-size in the $z$-direction is $\Delta z \sim $ 1m. Once again, this code seems to work correctly. To my understanding, explicit methods like the ones I've implemented and finite difference approximations are more accurate when using smaller step-sizes. Of course there should be a precision limit based on floating point error, but I can re-scale my variables to avoid this problem. Now I am using the exact same initial/boundary conditions as for the above, but wanting to simply consider the domain $t = [0,1000]$ and $z = [0,10^{-4}]$. It should correspond to my solution above but simply on a reduced domain, right? Well when I run the code now, $\Delta z \sim 10^{-7}$ and it appears there are very fast and large oscillations in only the $z$-component for $A, B,$ and $E$ (larger than any oscillations than on the original larger domain). I have an adaptive step based on the Cash-Karp method and the steps seem to work perfectly fine on this domain. But it appears my solutions on $t = [0,1000] \times z = [0,10^{-4}]$ and $t = [0,1000] \times z = [0,1000]$ do not correspond. When reducing the domain of $t$, the oscillations are still present, but just not as large. To my knowledge, it's arising because of the spatial derivative approximations where a small $\Delta z$ results in a large $\frac {\partial E}{\partial z}$. But is this to be expected when using such precise values with finite difference approximations? Are these large oscillations I am observing characteristics of the true solution or simply numerical errors? 
Limit to precision of step-size
finite difference;numerical analysis;error estimation;explicit methods