A Practical Way to Determine Time Step in Simulations
Hongbin Ju
Department of Mathematics
Florida State University, Tallahassee, FL.32306
www.aeroacoustics.info
Please send comments to: hju@math.fsu.edu
In numerical simulations, two
factors play key role on stability of calculations. One is time step ; the other is artificial damping. When instability occurs,
how to determine if the cause is too large
, or the improper (too small or too large) artificial damping?
Here is a practical way to figure it out.
Perform a series of calculations with smaller each time, and
record time steps where the calculation blows up. List the results in a table.
Here is an example from an actual simulation (
is the maximum
used):
nth Simulation |
|
|
(Timesteps at blowing up) |
|
1 |
|
|
700 |
|
2 |
|
0.9 |
2500 |
0.28 |
3 |
|
0.9 |
12600 |
0.2 |
4 |
|
0.9 |
14000 |
0.9 |
5 |
|
0.9 |
15500 |
0.9 |
6 |
|
0.9 |
17300 |
0.9 |
7 |
|
0.9 |
19200 |
0.9 |
For the first 3 simulations, the
time steps at blowing up increase drastically when reducing , which means
is the reason
for the instability. However, from the 4th simulation, the time steps at
blowing up only increase linearly with reducing
. That shows
is small enough
for the simulation. The instability is due to improper distribution of
artificial damping coefficients, which are either too small or too large. First
try larger artificial damping. Most of the time this will stabilize the
simulation. But occasionally the calculation may blow up even faster. Then one
may try smaller artificial damping. One of the two measures should make things
work.