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I am trying to determine the order of my numerical method for resolving a fluid-structure interaction problem using the immersed boundary method. I am using Crank-Nicolson to resolve the fluid equations and the immersed body is updated in time using an explicit operator splitting + projection method. The CFL condition for the fluid equations is

$$ CFL = |u_\max(t)| \frac{\Delta t}{\Delta x} \,. $$

I ran one set of simulations where I keep the timestep size fixed and refine the spatial resolution of the fluid and body meshes. This way, I'm able to determine the spatial order of accuracy of my scheme.

Now, when I try to keep my spatial resolution fixed and vary the timestep, I sometimes violate the CFL condition. Not wanting to resort to very minuscule timestep sizes, I did a convergence study where I held the CFL number fixed and varied the spatial and temporal resolutions proportionality. This tells me the overall order of my entire numerical method, however, does not distinguish between the spatial and temporal convergence rates. Is it true that in this case, if the temporal accuracy is less than the spatial accuracy, then the results will tell me the temporal convergence rate?

namu
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If you are trying to isolate spatial error, you will want to choose a fixed but very small time step so you can guarantee that temporal error is orders of magnitude smaller than spatial for every mesh considered. And obviously if you want to isolate temporal error, do the opposite, choose a very fine spatial discretization and vary the time step.

Truman Ellis
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  • Yes, this is true, however, I was asking a different question. Say you know that you have a higher order method in space compared to the time stepping method, and you have shown numerically that the spatial scheme achieve the desired order of accuracy. Now, if you vary both spatial and temporal resolution, such that the CFL is held constant, is the slope of the convergence study going to reflect the temporal order of the scheme? – namu Jul 14 '16 at 18:47