Hi,

Local time-stepping can be used to accelerate convergence towards a steady state solution. The steady state is reached when the diffence in values of two successive solutions is small. However, time steps being chosen locally, the time for the steady state solution then loses its physical meaning. The question I am asking is: Is it possible and useful to perform time accurate local time-stepping in transient computations? Does anybody know any references in regards to this matter?

Many thanks.

Pierre-Yves Lesage

Thinking about traveling into the future in real time? The local time-stepping is just a book keeping approach, something which can exist in your mind, that is the future, the present and the past all coexist in real time. In the real world, it probably don't exist. I don't know. It is hard for me to tell what is real and what is not real. "time accurate local time-stepping in transient computations" simply means that the future, the present and the past must coexist at the same time because of the definition of "local time-stepping"which allows different time steps to be taken locally at different point in space. Maybe I didn't quite understand your question.

One way to exploit "local" time stepping for transient cases is to use inner iterations. In other words, for each global time step (which is the same for all points in the field) you can perform inner iterations to eliminate factorization errors. Many times the inner iterations are performed in pseudo-time which accelerates the convergence to the "pseudo-time" steady-state for the current global time step. This approach is being used in the latest version of the WIND code being developed by AEDC and NASA Lewis (NPARC alliance).

As to your question about time-accurate local time stepping, I've never heard of it being done. The only way I can conceive you might do that would be to take large local time steps at the larger grid cells, hold them fixed in time, and compute the smaller grid cells at multiple smaller time steps until they catch up. That approach would quickly become a bookkeeping nightmare.

The standard adaptive mesh refinement (AMR) scheme of Berger and Oliger performs the scheme described by Doug. That is, it chops up the domain into grid cells, "halves" the ones that need refinement, and performs two timesteps on the halved cells for every one performed on the standard cells. The "halving" is continually performed until it reaches a converged solution, so one can end up with widely disparate grid cell sizes and, consequently, disparate timesteps at each cell. The scheme is explicit so global time accuracy is maintained.