CFD Online Discussion Forums - ADI of N-S equations

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 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.

regards

Dean

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.