Cats2D Multiphysics > Research Topics > Detecting temporal instability in time-dependent problems

All unpublished results shown here are Copyright © 2017–2019 Andrew Yeckel, all rights reserved

Detecting temporal instability in time-dependent problems

Widely separated time scales pose special challenges

I was doing some routine calculations of segregation in a vertical Bridgman system and found that a time-dependent oscillatory flow (a limit cycle) was lurking behind what appeared to be a perfectly normal quasi-steady flow. Let me explain what I mean by quasi-steady. In a batch system like vertical Bridgman, the geometry constantly changes as the melt is depleted, so system behavior is inherently time-dependent. But the growth rate is typically very slow, less than 1 cm/hr, whereas the flow can adjust on a time scale of a few seconds and the temperature on a time scale of a few minutes. Because the other fields respond so rapidly compared to the change in geometry, the time derivatives of these fields remain small and the solution at each time step very nearly satisfies the steady-state equations.

When this situation prevails the size of the time step is determined by the rate of change of geometry, which happens so slowly and smoothly that step sizes up to 20 minutes give very accurate results. I've tested this using the variable time stepping algorithm of Cats2D with three different error targets. In all cases the initial time step was 300 seconds. The plot on the left shows the time step size versus integration time for targets of 10-3 (green curve), 10-4 (red curve), and 10-5 (blue curve). The plot on the right shows the evolution of the average flow velocity over the course of the integration. The time step remains large and the time integration completely fails to identify the limit cycle, yet the average velocity is accurately calculated, especially for the smaller targets.

Gresho, Lee, and Sani (1979) recommend a target of 10-3 (0.1%) for the relative norm of the truncation error at each time step. This measure might seem small, but it represents only the local accuracy of the time step, not the cumulative error of the time integration. It was recommended at a time when computers were several orders of magnitude slower and coarse solutions were accepted as the best available. The default value used by Cats2D is 10-5, which reduces the time step by a factor of 4.6 compared to 10-3. Yet even this value failed to detect the limit cycle, because the time integration steps over the unstable period in the first hour of growth before the instability can grow.

To prevent this from happening, a smaller initial time step is needed. Reducing it from 300 seconds to 30 seconds has the effect shown below. Now the instability develops for error targets of 10-4 (red curve) and 10-5 (blue curve), but it takes longer to develop for the larger value, as seen in the plot on the right. At 10-3 the time step still grows too fast to detect the instability and steps over it.

Here are results for initial step size reduced to 10 seconds. Now the instability develops for all three error targets, but at 10-3 it develops more slowly, and is also not very accurate. One obvious conclusion is that 10-3 is simply too large for accurate work. But it is also apparent that simply using a smaller error target is not enough to guarantee capturing the true dynamics of a problem.

The situation posed here by these widely separated time scales is problematic. An initial step size of 300 seconds is perfectly reasonable, even a bit on the conservative side, for this problem, but to detect and accurately resolve the limit cycle requires a step size under 3 seconds. It demonstrates the constant need for care and vigilance when studying nonlinear systems.

I've seen this kind of behavior for two other problems, detached vertical Bridgman growth (DVB) and temperature gradient zone melting (TGZM). These systems all have in common a slow time scale for batch operation, and a fast time scale for a local instability. In DVB the capillary meniscus bridging the crystal to the ampoule wall becomes unstable. In TGZM the shape of the solid-liquid interface becomes unstable. In each of these cases it is possible to outrun the instability using a time step small enough to capture the batch dynamics accurately, but too large to resolve the faster time scale of the instability.