How can we define the precision we require in a numerical differential equation solver? What is it that I have to optimize to know? And how do I know that I'm at a sufficient time-step value?
For example, in those programs like Mathematica and Matlab, how does the differential equations solver know the suitable step size for your problem? How could that be automatically determined?
A full explanation of the approach would be nice (or maybe a term that can be googled).
What is typically done for ODEs is to solve at each timestep using two methods of different order. The discrepancies between the results are then used to estimate the error and then decide whether to decrease or increase the timestep length.
You can find more information on the subject here. Two examples of these types of schemes are RKF45 and Dormand-Prince.
for 1-D transport problem, under implicit method, we use Courant number, a dimensionless number to choose appropriate time steps:
$$\frac{u\Delta t}{\Delta x}\le C_{\max}$$
$C_{\max}$ should be less than 1, if we make it as 1, we get
$$\Delta t\le \frac{\Delta x}{u}$$
The basic idea is that, at one time step, a particle placed in the problem domain should never transport through two nodes where solving properties are defined.
for 2-D one can expand it as:
$$\frac{u_{x}\Delta t}{\Delta x}+\frac{u_{y}\Delta t}{\Delta y}\le C_{\max}$$
