[lzhaok6]

Dear all,

This might be a basic question but I haven't figured out an answer.

In explicit FVM or FEM simulation, the minimum global time step size is usually determined by the smallest element in the entire mesh. Thus, if several elements are extremely small compared to the other elements in the same mesh, the majority of element would run at a time step size much smaller than their stable time step size.

Now, I'm wondering if there is a scheme which allows the different elements to run at their biggest time step size thus save some computation. If not, why?

Thanks!

[agd]

What you are describing is referred to in the literature as local time stepping. It can be used for steady state calculations, as it destroys time accuracy. But for steady state calculations it can speed things up considerably.

[lzhaok6]

Originally Posted by agd

What you are describing is referred to in the literature as local time stepping. It can be used for steady state calculations, as it destroys time accuracy. But for steady state calculations it can speed things up considerably.

Thank you so much for your response! What about the explicit transient calculation? Is there any numerical scheme corresponds to local time stepping?

[agd]

Not that I am aware of. Local time stepping destroys the time accuracy of transient calculations. It may be possible to embed a type of local time stepping within a dual time step or Newton type iteration, but then you lose the speed of the explicit calculation and you might as well use an implicit scheme. Plus, you still aren't guaranteed that you would have time accuracy.

[praveen]

There are many papers on "time-accurate local time stepping". Just search online. Here are some recent ones

https://www.sciencedirect.com/scienc...09170815002390
https://www.sciencedirect.com/scienc...45782514002515

[agd]

Thanks for the heads-up. I'll take a look at these.

[lzhaok6]

Thank you all for your reply. Since I'm doing FEM modeling of the wave propagation problem using explicit solver. The following two papers are more relevant to me.

Grote, M. J., & Mitkova, T. (2013). High-order explicit local time-stepping methods for damped wave equations. Journal of Computational and Applied Mathematics, 239(1), 270–289. https://doi.org/10.1016/j.cam.2012.09.046

Grote, M. J., & Mitkova, T. Explicit local time-stepping methods for time-dependent wave propagation. Retrieved from https://arxiv.org/pdf/1205.0654.pdf

The basic idea is to divide the mesh into fine and coarse mesh region and integrate them in time separately. It's like coupling two spatial regions in the time domain.

However, I'm not sure if this approach is robust since little application is found. No commercial solver supports Local time stepping (LTS) to my knowledge.

[LuckyTran]

In steady state solvers this is used to accelerate convergence.

For time-accurate transient cases, I'm not aware of it being used in any commercial code. Robustness is most likely the reason why you don't find it here.

I know of at least one example off the top of my head: an acoustic solver at FSU which uses what they call multi-size multi-time method.