ODE45: small time-step size

Hi,

I'm trying to solve the transient 1D heat diffusion equation (dT/dt = a*d2T/dx2) with ODE45.

If I impose a short time interval (1000seconds with Dt=10s) everything is fine.

But if I want to solve the system on a long time interval (20000s), the solver starts to take very very small time step sizes when solving at t=±2500s...

I read in one of the previous posts ( ODE45 errer :input argument y is undefined ) that Jan Simon refers to such problems. I quote him:

"Btw., the function to be integrated must be smooth, otherwise the step-size control of ODE45 can lead to unexpected effects: [...] ODE45 reduces the step size to such a tiny value, that the integration takes hours to run and the accumulated rounding error dominates the solution. This is exactly what the step size control should avoid."

I think this is exactly my problem.

By reading some of his recommended references in the same post, I tried to run it using ode15s, but the same issue occurs at around t=±2800s and after a few iterations I get the error message "Unable to meet integration tolerances without reducing the step size below the smallest value allowed at time t".

So, I don't know how to solve this problem.

Thanks for your help

Shayan

Answer 1

[Mischa Kim]

Shayan, as indicated in the equation above the heat equation is a PDE (as opposed to ODE). Check out the documentation and examples on how to solve the heat equation.

[Shayan]

Thanks. I'm going to read it, but the thing is that I discretized my equations so I have a nonlinear ODE system.

[Mischa Kim]

I see. Is the heat equation the way you set it up supposed to be a stable system? At any rate the documentation and examples linked above should be a good reference solution you can work of.

[Shayan]

I discretized it using 2nd order central differences... I'm not sure but I think it should be stable...

Answer 2

[Jan]

The problems I've mentioned concern discontinuities only. The step size control cannot handles them. "Fortunately" it reduces the step size until the discontinuity is covered by the round-off errors and the intergration "works" at least. But the results are not trustworthy anymore due to the huge number of integrations steps and the accumulated round-off errors.

But the integration by ODE45 can suffer from otehr problems also, most of all the stiffness and instabilities of the problem. In the first case a solver for stiff problems can help, in the second case another mathematical method is required.

Answer 3

[alexander]

Hi,

I'm having a similar issue where the step size reduces significantly and then produces an error. The error occurs when running a simulation on my laptop using R2011a. When I try the same model on R2014a there is no issue so I don't believe it is the model. The simulation has a repetitive step in which the system recovers after each step, the error only occurs after several steps have been completed.

Could this problem be due to a system hardware issue, perhaps running out of memory? I am running 64-bit Windows 7 on an Intel Core i3 CPU M370 @ 2.40GHz with 4GB of RAM.

I have seen a similar issue before that has worked on an identical (factory spec) laptop but not on mine.

Thanks

Answer 4

[Shayan]

Thanks for the replies.

My BCs are: - imposed heat flux, dT/dx=q, on one side - adiabatic wall on the other side, dT/dx=q=0

Note that I also tried the 'pdepe' function. It turned out that the result are pretty close to my discretization/ODE method (for span times smaller than the limit after which the issue appears). It is much more faster as well (about one hour for ODE, while PDEPE finishes in a couple of minutes). However, the same problem persists...

Any idea please ?

Thanks.