Time stepping in comsol multiphysics

Question

I would like to know which is the algorithm Comsol uses in order to correct the time step it uses. For example, when you try to solve an equation you've written in the PDE coefficient interface, and you set free time-step, Comsol automatically choose the time step in order to obtain the specified absolute error. Anyway, how does Comsol choose the correct time step?

• In Navier Stokes equations the parameter to look to, is the CFL number, that has to be smaller than one.
• For generic transport problem the parameter is the Peclét number.

But what about all the other problems? There is a general way to find a toughtful maximum time step?

Answer 1

Given a PDE choosing the correct numerical solving strategy requires some knowledge/expertise. (Computational Science is indeed a "Science" and one has to learn it.)

In application specific software (e.g commercial FEM solvers for engineering problems like crash problems or metal forming) this knowledge is somehow "crystalized" and embedded, so that most application engineers should be able to numerically solve physical problems without too much trouble. Analysts can concentrate on the physics of the problem, and not too much on the numerics. This is an optimistic view: really tough problems require multidisciplinary expertise (num. analysis, computer science, engineering, physics) even when application specific commercial software is used; but at least, in a day by day use, big errors do not occur.

On the contrary your question is very general and broad, asking for guidelines applicable to a generic PDE and solving strategy, in a non-application specific environment.

I think that without any little hint to the type of problem you are trying to solve it is impossible to give you a sensible answer, as it is impossible to condense a sizable chunk of Computational Science in a few paragraphs.

Nevertheless I think that some useful advice could be given.

1. Physical quantities have dimensions: saying that $10^{-12}$ is a very small absolute error is meaningless. Check your equation for dimensional consistency; if possible try to write them in a non-dimensional well-scaled form.

2. For PDE's the "correct" time step may depend not only on how the equations are scaled (choice of physical units of measure) but also on the spatial discretization (finer mesh, smaller time step.) If you do not have theoretical results at hand, compare results varying both time-step and mesh-size.

3. Always try to compute a global quantity (e.g. kinetic energy) and plot its evolution over time at variable time steps: you will gain insight on what's going on.

4. There is nothing wrong in a trial-an-error procedure, provided you are able to asses if a given solution is physically meaningful.

Not much help, I know, but I hope this will be a good starting point.

Answer 2

It's not possible to answer your question in general, since comsol likely uses different methods depending on the problem.

The general answer is that a partial differential equation is discretized in both space and time. Just like you have to set up a volume grid, you have to break time into chunks. The solution accuracy is a function of the time-step "grid".

What variable time-step algorithms do is try to dynamically pick the time-step, to make it smaller when the solution is changing rapidly. This can involve testing a bunch of different time-steps, and picking one that gives acceptable error with respect to the finest grid it's checked.

The maximum time-step then controls how quickly the solution can be found when the solution looks linear in time. It should be on order of the relevant time-scale of your system, since if your estimate is way off, the integrator won't be able to draw a smooth line between its attempted time-steps. This may happen, in particular, if the thing oscillates - in which case you'd need less than the oscillation period.