A Practical Way to Determine Time Step in Simulations

Hongbin Ju

Department of Mathematics

Florida State University, Tallahassee, FL.32306

www.aeroacoustics.info

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.