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Vasiliy January 6, 2012 10:47

Implicit method for Navier Stokes equations
Hi All,

I construct implicit solver for Navier Stokes equations.
I use Roe method to calculate explicit fluxes.

My question is about convective and viscous Jacobians in implicit part.
I was told that some of such Jacobians can be excluded from large sparse matrix to accelerate convergence.

It is not connected with technic where viscous Jacobian is changed with spectral radius.

It looks like some of Jacobians are fully removed to accelerate convergence.

Does anybody have information about such algorithms?


agd January 6, 2012 13:41

Are you referring to something like this?

If so, try googling "pulliam diagonalized". Thomas Pulliam is the guy who has put a lot of developmental effort into such schemes. You should be able to find quite a bit of information on the internet.

Martin Hegedus January 6, 2012 16:50

Are you referring to implicit calculations for structured grids? I don't consider myself an expert, but here is how I see it.

There are four AF (approximate factorization) groups. The idea of an AF method is to simplify the matrix inversion by means of approximations.
ADI (Alternating Direction Implicit)
DADI (Diagonalized Alternating Direction Implicit) The Pulliam Chaussee method is in this class
DDADI (Diagonal Dominant ADI)
D3ADI (Diagonalized ADI)

These methods are explained in "Convernce Characteristics of Approximate Factorization Methods" T.H. Pulliam, R.W. MacCormack, and S. Venkateswaren. I've not seen a free version of this paper on the net, in which case you'll need to pick it up from your local university library.

The idea behind DADI is to solve for the eigenvalues of the Euler Jacobians. This creates a pentadiagonal matrix which is faster to solve than the tridiagonal block matrix solver required by the ADI method. However, due to factorization error, the DADI method has a CFL penalty compared to the ADI method for certain grid topologies and flow conditions.

The DADI is also not time accurate, so it should be used with a subiteration approach for time accuracy. However, IMO, the ADI method should also be used with a subiteration approach for time accuracy.

Both the ADI and DADI are not diagonally dominant for large CFL or dt numbers, therefore they will bomb with very large CFL values. Usually a CFL between 10 and 20 gives you max convergence rate.

In theory, any CFL value can be used with the DDADI and D3ADI methods. However, larger CFL values do not necessarily mean faster convergence rates.

klik24 January 12, 2012 08:43

Help please
Hi Vasily,

I am developing the same code for unstructured grids. Can you help me about the implementation of Boundary conditions :confused: ? (Any paper or book would do) I think I did not look properly.

Thank you for your help.

Sorry about asking a question for your question.

Abhishek :)

vinayender January 12, 2012 09:05

Refer Yoon Jameson Paper
Vasiliy ,

Please refer "AIAA 86 0105"
"LU Implicit Schemes with Multiple Grids for the Euler Equations"
A.Jameson and S.Yoon
Princeton Univ., Princeton, NJ


for Implicit equation, of form
(1/del_t - del_R/del_U) delt_U ) = R(U_n)
Right hand side is imprortant because it is the governing physiscs and for steady problems LHS is just numerics. so its accuracy is not important.
so for LHS you use only a very quick schems which refered in the paper said above......refer it

vinayender January 12, 2012 09:11

if you are implementing a density based solver, you can refer "Computational Gas Dynamics " by Laney

and also a book "Computational Fluid dynamics: principles and applications" by Jiri Blazek

klik24 January 12, 2012 09:32

Dear Vinay,

I have already finished coding the entire Implicit algorithm for unstructured grids based on Roe scheme (upwind). But I am having a difficulty applying boundary conditions (to be specific calculating Convective Flux Jacobian at boundaries specifically Wall, Symmetry) in the implicit operator. I am trying to solve the system using a Newtons method. My code is based on an explicit code I previously developed. Any paper or book suggestion is highly appreciated.

Kind Regards,

vinayender January 13, 2012 01:45

Ghost Cell or Half Volume construction
Hi Abhishek,

There are few procedures to calculate the fluxes on wall (same with little different for symmetric bc)
1) construction of ghost cell

1) Construction of ghost cell, you can assume a ghost cell in the wall region of same volume jsut opposite to the real cell and define the properties of that cell as same values other than for velocities. velocity compenets can be calculated such that the net mass flux at the wall ( the interior face between real cell and ghost cell) is zero..
values of real cell at wall can be rho, p, temperature, u and v
values of ghost cell at the wall can be rho, p, temperature, u_ghost and v_ghost
u_ghost and v_ghost can be calculated using 2 relations. 1 is normal mass flux is zero and 2nd is wall tangential velocity is 0 for wall and tangential velocities a finite one for symmetric bc)
Now flux at wall can be caculated at the face (which is between real and ghost cell) like you calculate for any other interior face......

vinayender January 13, 2012 01:50

got wrong ?
Hi Abhishek,

i guess i got your question wrong.. let me get back to you

Vasiliy December 3, 2012 14:07

Hi Everybody,

Thanks for previous help.
I have implemented Weiss and Smith precondition.
It works fine for laminar flows but I have slow convergence for turbulent flows.
It looks like there is method to improve convergence of turbulent flows. I saw it in some hard code results (Fluent for example). My current task is to minimize number of Navier-Stokes iterations.
Does anybody know such methods?


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