ADI of NS equations
Dear all:
Could anyone tell me how to deal with nolinear terms in NS equations with ADI method and the pressure terms? Thanks in advance. Zhou Yongcheng 
Re: ADI of NS equations
(1). The book by Anderson, Tannehill and Pletcher, "Computational Fluid Mechanics and Heat Transfer" has a chapter on numerical method of NavierStokes, which includes ADI related schemes. (2). You can read the text and find the reference materials.

Re: ADI of NS equations
Suggest add to it the other book 'Applied Numerical Methods in Engineering' by Carnahan, Wilkes and Luther.

Re: ADI of NS equations
First I want to say that , if you are using ADI for 2D NS equations then it is ok. For 3D NS equations you better go for LU Approximations(example: LUSGS by A.Jameson).
1)We will treat Nonlinear viscous terms explicitly. 2)pressure term will be expressed in terms of u,v,w,rho and e. 
Re: ADI of NS equations (Paper ref.e w. Abstract)
Hi,Yongcheng
I published a conference paper on this topic last year. A starting link can be found at: http://edwww.lerc.nasa.gov/thermal/t...urses.html#PAP The title is: On the Adaptation of the ADIBrian Method to Solve the AdvectionDiffusion Transport Equation I grabed the abstract from the file for your review. See below. The nonlinear terms are linearized by belating 1 time step, the leading velocity term. The advection terms are split in a fashion analogous to the diffusion terms by following the Brian adaption. To my knowledge this had not been done before. The pressure term is simply a source to the AD equation. The ADIBrian routine applies 3sweeps and is 3D. The third dimension is no problem. I think the performance was excellent. It outpaces SOR by x30 and this was for a 2D problem in which I was executing the 3rd sweep unnecessarily. Hope this helps. regards Dean ABSTRACT: The focus of the present study is a semidirect solution to the linearized Burger's advectiondiffusion (AD)equation using alternating direction implicit (ADI) methods. In particular, the paper features the adaptation of the Brian ADI method, originally designed for stable three dimensional (3D) solutions of the parabolic heat equation, to include the advection component of the Burgers equation. The present study presents a method to split up the advection component in a manner which is consistent with the splitting of the diffusive terms in the Brian method. Upon implementing upwind differencing, this new method offers very robust stability margins and is capable of issuing stable solutions at Courant numbers exceeding 10. The upwind scheme applies only the left or right diagonals of the ADI coefficient matrix to register the advection term depending on the direction of the velocity vector. For this reason, upwind differencing is an ideal starting point for the ADI solution method because ADI methods depend on a direct inversion of a tridiagonal coefficient matrix. However, for large Peclet numbers, the advection term dominates the diffusion term in the Burgers equation and the solution is hampered by the classical numerical diffusion induced by upwind differencing. This motivates the search for enhanced differencing schemes which can be implemented with the ADI method. A central differencing scheme produces secondorder spatial accuracy and can be differenced within the tridiagonal band and eliminates numerical diffusion, but generates dispersion errors. To mitigate both diffusion and dispersion errors, thirdorder upwind differencing is implemented. Thirdorder upwinding requires four points (i  2, i  1, i, i + 1). In the tridiagonally bound ADI method, the fourth point (i  2) is registered as a source term using the belated ADI state. Effectively, the thirdorder upwinding is implemented as either central differencing with a smoother or upwind differencing with a sharpener. Both give the same numerical results. All three advection differencing methods are compared to a showcase of steady and transient exact solutions to the Burgers equation which demonstrates the combined utility of the new advection method with an ADI solution engine. 
Re: ADI of NS equations
Upon reviewing my archives, I've found this image:
http://www.ctacourse.homepage.com/Ga...FrontQuick.jpg This was derived with the ADIBrian simulation and is a simulation of the AD equation with infinite Peclet number. Flat front simulation. The solution is derived using the modulator function as discussed in my abstract. It clearly shows the mitigation of both diffusion and dispersion errors. I can't say enough about ADI. It is an excellent adaptable method, having even used it in phase change heat transfer simulations. regards Dean 
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