Hi,

I am working on higher order methods to solve euler equations. I have recently extended the code to curvilinear coordinates. The code works fine. But, I noticed that, the calculation of jacobian transformation matrix is not very accurate and this leads to spurious oscillations in the solution.

As of now, I dont know how to calculate the Jacobians accurately so that the error is reduced during transformations...

When the grid lines are straight, the error is quite negligible. But, when I have a grid like x=u, y=v+sin(u) (u,v are from 0 to 1), I have some error terms. I cant reduce the error even if I increase the order of accuracy to 8. If anyone is aware of such problems, please guide me.

Regards, DSS

If you don't mind, can I see yours in detail? I'd like to take a look at it and let's see how to help you.

I will try to explain what I did to quantify the errors.

I generated the grid points according to the equations, or by solving elliptic equations.

Now, I calculate the Jacobian matrix ( d(xi)/d(x) ) where xi is the computational coordinate and x is the physical coordinate.

With the available transformations, I solved the standard equation

d(Q)/d(t) + d(F)/d(x) = 0

in the transformed sense. I used the conservative form so that the fluxes become

Ft(xi) = 1/J ( F_x d(xi)/d(x) + F_y d(xi)/d(y) + F_z d(xi)/d(z) )

Now, ideally, when I solve this with F=1 for all values, I should get d(Q)/d(t) = 0. This, I am able to get to the value of 1e-13 if the grid lines are linear. However, when the grid is skewed, this error increases to the order of 1e-3.

I calculate the derivatives using a 5 point FD stencil, which transforms to backward and forward differencing near the boundaries. I witnessed the same with 3 point and 9 point stencils.

As mentioned before, I used the grid of the sort

x = u y = v + sin(u)

where (u,v) range from 0 to 1.

Hope this gives enough information.

What is the grid spacing you use for the computation.

Used 51 points for each direction for the equation

x = u y = v + sin(u)

u,v from 0 to 1.

The error is more as you move along x axis.

Did you check the value of jacobian at each point.does any point have jacobians close to 0.Also how does the derivatives (dxi/dx) vary.Do they have some sort of steep rise in some regions.Abrupt variations in metrics can produce parasite waves.

actually, the problem is not abrupt variation of the metric values. But, the inability to calculate the derivatives d(x)/d(xi) accurately, leading to error in calculation of d(xi)/d(x) and the Jacobian. This results in loss of conservation in the transformed equations.

So, when you start the iteration, there are some parasitic waves which are visible if you do a low mach no. case.

actually, the problem is not abrupt variation of the metric values. But, the inability to calculate the derivatives d(x)/d(xi) accurately, leading to error in calculation of d(xi)/d(x) and the Jacobian. This results in loss of conservation in the transformed equations. So, when you start the iteration, there are some parasitic waves which are visible if you do a low mach no. case.

Generally you do not want to calculate the metrics accurately but to arrange the differencing to numerically conserve what is important in your simulation. You can test this by using analytical metrics and comparing with your optimum arrangement of differencing terms.

There were quite a few papers on this subject in the 1970s and early 1980s when the use of curvilinear coordinates started to be heavily used for CFD but I am afraid I have no references to hand.

I think Andy is correct. I don't have any references that deal specifically with this issue but I know that you can find them in Yusuf Ozyoruks Ph.D. thesis (The Pennsylvania State University). He implemented a higher order method to solve the Euler/NS equations for the purpose of predicting fan noise. You might be able to download it from Lyle Longs Website. (It appears to be down right now though).

The classic reference is an AIAA paper by Thomas and Lombard on geometric conservation.