0

Is it possible that a particular scheme (MAC in this case) gives accurate results for one grid size but inaccurate/no results for finer grid size or finer mesh? If yes, then how or why?

learntocode
  • 9
  • 2

1 Answers1

2

Short answer

Yes.

Longer answer

For a general question, a general answer: numerical procedures for solving partial differential equations have stability criteria, which usually link the time step dt to the spatial step dx. Usually you have to make sure that dt is small enough for a given value of dx, otherwise your procedure will be erroneous or even diverging.

After googling for "fluid dynamics marker and cell stability" I found a review about the MAC method [Computers & Fluids 37, 907 (2008)] in which Eq. (42) states that dt<dx/norm(u), and there's also Eq. (43), which might or might not be the actual stability criteria for the MAC method (but I'm unfamiliar with the particular method and didn't feel like reading through the paper, just took a glimpse).

For your specific scheme you have to find out/derive the stability criteria, then make sure to abide by them all. Only then can you start to expect that the output has any resemblance to the actual solution of your PDE.

Andras Deak -- Слава Україні
  • 33,737
  • 11
  • 83
  • 111
  • I changed the time step to minimum possible level: 10E-8. But then also, the results were spurious. I was just wondering if its possible that a given scheme general , the stability criteria is a function of time step and spatial step (dx) only or any other variable might also be there? – learntocode Sep 25 '15 at 15:59
  • @learntocode I still feel reluctant about trying to answer questions this broad. Yes, other variables **will** be there. For instance, `dt` and `dx` have different dimensions. You need other factors to get a meaningful relationship. These are definitely related to your differential equations, and possibly to other stuff (like your geometry, it might matter if your system is of length `L`. There are many kinds of numerical schemes. Who knows? I certainly don't. So how about asking this on [math.SE](http://math.stackexchange.com/)?) – Andras Deak -- Слава Україні Sep 26 '15 at 23:06