Presence of multiple spatial and temporal scales in the same physical system

Question

How does the presence of multiple spatial and temporal scales present in a physical system make it difficult to numerically simulate the system.

Answer 1

Suppose I have a signal that is a sine wave of frequency of 1 Hz and I want to show you a discrete sampling of that signal. Let's say to get a good representation we agree that I need $A=10$ data points per period. If I want to show you one period then I need 10 data points. If I want to show you $B=1000$ periods then I need 10,000 data points.

In the case with one period there was only one time scale involved, the period of the signal was $T_{per}$ and the length of the window was $T_{tot}$ but these were equal so we only needed $A$ data points.

In the case with 1,000 periods there were two time scales involved, the period of the signal $T_{per}$ and the total window length $T_{tot}=B T_{per}$. We see that we needed $BA$ data points.

Here $A$ is something like a fidelity factor, $A$ is higher if you want a better representation of the signal. $B$ is a seperation of time scales factor. It is the ratio between the shortest and longest timescale you would like to represent.

Another example, suppose we have a signal which is composed of a signal at 1 Hz ($T_{1,per} = 1\text{ s}$) and a signal at 1 MHz ($T_{2,per} = 1 \text{ $\mu$s}$). If we keep the same fidelity factor of $A=10$ then that means our samples must be spaced in $100 \text{ ns}$ intervals, but if we want to represent the 1 Hz signal we need to at least have $T_{tot}=1s$ meaning we must have $10^7$ samples. Here $B=10^6$.

So we see that a separation of timescales is difficult to simulate because the high frequency signal forces you to have high frequency sampling but the low frequency sampling forces you to have a very long signal. These two effects result in a very large number of samples necessary which uses a lot of memory and makes computation very slow.