A 2D test case for Linear reconstruction FV scheme?
 

Hello,

Does anybody know a 2D test case for accuracy study of a linear reconstruction Finite Volume or maybe the MUSCL scheme?
I would prefer a test case without periodic boundary conditions.

Many thanks,
Faraz

You could test a linear solution of Df/Dt=0 in a 2D velocity filed u=-omega*y, v= omega*x which is a rigid rotation. Set some initial field f(x,y,0), after one complete rotation the solution should be equal to the initial state

Thanks,

That is a very interesting example. Actually I did something similar to that, but within the domain x = [-1,1] & y = [0,1] for which there seems to be a singularity of velocity at (x,y) = (0,0).

However, I did another example for an angular convection:

x = [0,1], y = [0,1],

and veloctiy: u = Omega, v = Omega

So, it converges with correct rate for N = 8, 16, 32, ..., 512 (conv rate = 2 for linear)

But after N = 512, 1024 the convergence rate degrades for some reason (conv rate = 1.7 e.g. for linear), which is quite strange ! and it is not due to zero error space because the error is still 1e-6 or so.

but u=v=constant is a uniform translation velocity field ...

That's true, but would it cause a problem?

what about the initial condition and BCs ?

Well, for angular convection, the initial profile at inflow ghost cell is either a Gaussian or a sine/cosine distribution on ghost cells adjacent to either x = 0 and y =0, and zero everywhere else.

Regarding BC, for left state of the face at the inflow BC, an average of the ghost and the adjacent internal cell is performed. Because with the constant reconstruction at the inflow BC, the convergence gets close to 1st order other than 2nd order.

Is there a better boundary treatment for the inflow?
I also noticed that the convergence rate is a bit dependent on initial profile !

x=0 and y=0 are inflow boundaries where you prescribe Dirichlet BC, x=1 and y=1 are outflow boundaries where you can prescribe homogeneous Neumann BC. What you can do is to run in time until a steady state (for steady inflow) is reached. Then compute the errors and repeat for via via refined grids.
Use the same dt for all run, take value that allow a stable computation on the most refined grid.

Yes indeed, but unfortunately the same degradation of order in higher number of cells remains. But, as I mentioned, it is also a bit dependent on the initial profile I define on the Dirichlet BC.

Apart from this problem, do you think if with the rigid rotation example I should obtain the right order of convergence (in spite of that singularity in (0,0) )? Do you, by any chance, know any references to check with that?

I really can't find any relevant examples in the literature or papers.

Thanks again.

the initial and boundary conditions must be smooth to check the correct order of accuracy. I remember several tests in journals, check in J.Comp.Phys as well as in Computer & FLuids, Int.J. Num. Meth.FLuids.

Have also a look at the book of LeVeque

Thanks, and if somebody already knows any refs, please let me know. Thanks