The accuracy and precision of a numerical simulation are highly dependent on the resolution
of the space-time grid. In order to keep the users from shooting themselves in the foot,
we recalculate these parameters whenever a significant change occurs in the system.
For example, the time step is reset whenever the
equation type is changed or a segment is created, destroyed, or moved.
The default value of the time step, 
, is equal to 
of the
maximum stable value for
the diffusion and Schrödinger equations and is equal to 
for Klein-Gordon
type equations.
where 
and 
are the left and right boundaries of the medium
and N is the number of points.
Describe the
relationship between number of points and accuracy of the simulation.
Is the algorithm first-order or second-order in the number of points? e.g. What happens to
the error when the number of points is cut in half? How does
computation time depend on the number of points?
by a factor
of 2 above the default value. Use the [F4]
hot key to single step
the algorithm. Clearly our algorithm has become unstable.
noted in the previous exercise. A good way to check for these types of
errors is to increase the number of grid points or to decrease the time step (or both!)
and see if the character of the solution changes.
Use default values for the number of grid points for the following exercises.
; it may not be accurate.
Examine the behavior of the diffusion algorithm with large time steps. Change the initial
disturbance to a Gaussian pulse after selecting the equation type. Use
[Parameters]|[Time Parameters]
and increase 
by a factor
of 100 above the default value. Run the simulation and notice the non-physical
behavior of the simulation.
, enter 
into the text field of the [Medium]|[Parser]
input screen. Select a Gaussian initial
wavefunction having no average momentum, i.e., k=0, but offset from the equilibrium point.
Set the boundary condition to periodic in order to show momentum eigenstates.
Run the program with analysis set to FFT of 
and contour plots. Run the
simulation and notice the spurious wavefunction. How must the system parameters be changed
to accurately model the dynamics of this system?