# Time iteration no longer smooth after using scaled units

I have a time iteration function looked on a 2D surface like this. Since the numbers wee very small i.e. hbar=6.6260700404e-34./(2*pi), my professor told me to use our own "scaled unites" during the calculation. ie. instead of using $1e-9$ (meter), scale the unit and use $1e-6$ ($\mu m$) during the calculation. e.t.c.

It worked very well when I was only changing the space parameter, (m) However, when I adjust the space scale and time scale together. I encounter a problem, where the function stopped being evolving smoothly. I checked my substitution, and they were correct. But what happened here? Can we really save time of calculation by using scaled units?

• If you're solving Schrödinger equation, I suggest that you use a system of equations were constants like $h$, $m$, and $e$ are all unitary. – nicoguaro Apr 8 '18 at 2:46
• I think that your marching time method becomes unstable. – HBR Apr 8 '18 at 15:51
• @HBR Why it became unstable? – J C Apr 8 '18 at 16:32
• These peaks result when some iterative method becomes unstable, like when using euler forwards scheme the Courant number is more than 1 for a pure convection equation. Be sure that this number (ratio between $\Delta t$ and \delta x\$) is under 1. – HBR Apr 8 '18 at 16:35
• And for the last question. No. Scaling your nondimensional numbers allows you to control better the numerical aspects. It is always better to treat with numbers of the order of unity. – HBR Apr 8 '18 at 16:53

These peaks result when some iterative method becomes unstable, like when using euler forwards scheme the Courant number defined as: $$C = \frac{v \Delta t}{\Delta x}$$ where $$\Delta x/v$$ must be the minimum residence time in a computation cell defined as the characteristic length of this cell $$\Delta x$$ divided by its local velocity $$v$$ and this number must be compared with the simulation timestep $$\Delta t$$. For an explicit time-scheme the Courant number is usually under 1 for a pure convection equation.