Dear Friends,

My question is with regard to order of accuracy. I have solved two inviscid flow problems

1. Euler flow past a cylinder at M=0.3, Solid wall BC on cylinder implemented as pressure extrapolation bc and Reimann characteristic bc at the farfield.

2. Supersonic vortex problem, where v~1/r in between two concentric circular arcs( Aftosmis, Gaitonde and Tavares: AIAA Jl, 33(11) ), Wall BC as above at the inner and outer arcs, Supersonic inlet and outlet bc, with the velocity at the inlet varying inversely as the radius.

I have made use of a linear reconstruction procedure with Roe's scheme in both cases. These problems have an exact solution available, so I compute the error in the domain ( Error in i th cell = Exact solutio at i - Numerical solution at i).

The plot of L1 norm of error with the number of volumes in both cases shows an error fall rate close to unity. The problem is that while the fall is expected, the rate for problem 2 as in Aftosmis etal. is 2 instead of 1. The same rate could logically be expected for problem 1 but I still get a fall rate of 1 instead of 2.

I am not able to figure out where the mistake is, because I have the solution right. I understand that there are methods like DGM which are sensitive to BCs and which could affect the fall rate to a greater extent, but I do not think it is the case with FVM. As far as grids are concerned, the first problem is evaluated on progrssively refined structured grid and the second on regular quads of near unit aspect ratio as well as unstrucutred grids.

Any suggestions are most welcome

Thanks in advance,

Ganesh

What is your definition of "L1 norm of error" ?

I would compute the L1 error as

Σ_i | U_i - u(x_i) | A_i

where U is the numerical solution and u is the exact solution, with x_i being the centroid of the i'th cell. This is a discrete approximation of the correct integral definition of L1 norm. Note that it may be more accurate to take the linear reconstruction within each cell and compute the L1 norm of the error. Plot this error against a grid size, for example,

h = max_i sqrt(A_i)

There is no exact Euler solution for Mach 0.3 flow over cylinder. Your are probably using the potential solution.

Dear Praveen,

The definition of my L1 norm of error is as you have mentioned in your post. I have plotted this error against the square root of number of volumes(or nodes) or against a characteristic grid length, like sqrt(area) in cases where it may be available, like in cartesian grids. Also, I make use of the potential solution for comparison, Euler at Mach 0.3 is just the compressible(incompressible) approximation to that flow problem.

Thus, in either the cylinder case or the vortex problem, the slope fall is 1 instead of 2. I have observed the fall rate of 2 for linear scalar problems, like circular advection and this has given rise to a confusion.

Regards

Ganesh

I too have seen second order behaviour for scalar problems. But I have not done any tests for Euler equations. Limiters can easily degrade the accuracy. Are you using limiters ? If yes which one ? How fine is the grid ? The order of accuracy is an asymptotic property; the grid has to be sufficiently fine before you observe the correct error decay rate. Another possibility is the presence of singularities like corners which can degrade the accuracy. Finally, what about boundary conditions ? It is usually said that the boundary treatement can be one order less than the interior scheme but this may not hold in general. You could use the exact BC to eliminate this possibility.

Using sqrt(n) is fine for cartesian grids but not for unstructured. It is better to use a characteristic grid length scale.

Dear Praveen,

Thanks for the comments. I agree that limiters do degrade the accuracy, but the extent of degradation is not ver severe in terms of the global error fall rate as reported in Aftosmis etal. In any case, I do not use limiters for either of the problems, as my interest is in smooth continuous solutions. It is true that the order of accuracy is an asymptotic property and that we have to be in the asymptotic radius to see the true fall rate. These problems have been benchmarked by Aftosmis and Berger etal. and they have seen the fall rate on grids with lesser number of volumes, under the same solver conditions. These problem are also devoid of any singularities that could degenerate accuracy. As you have mentioned, I think the possibility could be in the BCs, however I am not sure if FVM has such a huge sensitivity to BCs, like other methods such as DGM. In any case, I will try the suggestion of having exact BCs.

In all my cases, I have made use of a characteristic length scale with structured grids, but in unstructured grids there are literature( Aftosmis, Berger et.al.) where they have used the number of volumes as also the sqrt(max(Ai)). I believe either would do the trick, if not could you tell me why the number of cell volumes is not a goof choice for unstructured meshes?

Thanks and Regards,

Ganesh

Instead of comparing with teh potentail solution why don't you estimate the order using three grids with constant refining (i.e every grid twice as fine as the previous, defining a mesh size as you wish) See Roache, or book by Ferziger & Peric. How is the error fall in that case?

In a non-linear problem with a second order scheme you are never going to get second order fall of thr error (ther is no perfect world !!)

If you refine cartesian grid uniformly, say double the grid points in each direction, then the grid size h is proportional to 1/IMAX (and also to 1/JMAX) and is also proportional to 1/sqrt(IMAX*JMAX) (I am assuming 2-D situation. You can see how you would have to change it for 3-D). On unstructured grids it is not clear how the grid size varies with the number of cells. It of course depends on how you refine the cells. Suppose you have a nearly isotropic triangulation and you refine the triangles so that they remain nearly isotropic then you can find some relationship between h and the number of cells. Otherwise the number of cells may not give a good indication of h.

Dear Salvador and Praveen,

Thanks for the comments. I am making an isotropic refinement of the domain and so the number of volumes could give a good picture. Although not a perfect world, a linear reconstruction is supposed to give a near second order accuracy as literature shows. It is a nice idea to check using Richardson estimates, but I still do not know what causes the fall rate to be 1 for the supersonic vortex problem, which has an anlaytical solution, for which Berger and Aftosmis report a global error fall of 2.

Thanks and Regards,

Ganesh