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Zhou Yongcheng January 7, 2001 22:40

ADI of N-S equations
Dear all:

Could anyone tell me how to deal with nolinear terms in N-S equations with ADI method and the pressure terms?

Thanks in advance.

Zhou Yongcheng

John C. Chien January 8, 2001 20:02

Re: ADI of N-S equations
(1). The book by Anderson, Tannehill and Pletcher, "Computational Fluid Mechanics and Heat Transfer" has a chapter on numerical method of Navier-Stokes, which includes ADI related schemes. (2). You can read the text and find the reference materials.

Mukhopadhyay January 8, 2001 23:19

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

A. Rajani Kumar January 9, 2001 05:29

Re: ADI of N-S equations
First I want to say that , if you are using ADI for 2D N-S equations then it is ok. For 3D N-S equations you better go for LU Approximations(example: LU-SGS by A.Jameson).

1)We will treat Non-linear viscous terms explicitly. 2)pressure term will be expressed in terms of u,v,w,rho and e.

Dean Schrage January 17, 2001 12:04

Re: ADI of N-S equations (Paper ref.e w. Abstract)

I published a conference paper on this topic last year. A starting link can be found at:

The title is: On the Adaptation of the ADI-Brian Method to Solve the Advection-Diffusion Transport Equation

I grabed the abstract from the file for your review. See below. The non-linear 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 ADI-Brian routine applies 3-sweeps 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.



ABSTRACT:  The focus of the present study is a semi-direct solution to the linearized Burger's advection-diffusion (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 tri-diagonal 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 second-order spatial accuracy and can be differenced within the tri-diagonal band and eliminates numerical diffusion, but generates dispersion errors. To mitigate both diffusion and dispersion errors, third-order upwind differencing is implemented. Third-order upwinding requires four points (i - 2, i - 1, i, i + 1). In the tri-diagonally bound ADI method, the fourth point (i - 2) is registered as a source term using the belated ADI state. Effectively, the third-order 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.

Dean Schrage January 18, 2001 08:58

Re: ADI of N-S equations
Upon reviewing my archives, I've found this image:

This was derived with the ADI-Brian 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.



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