Hello Folks, I am writing an implicit solver that solves the transonic flow over an airfoil. The scheme I'm trying to use involves the solution of a quadradiagonal set of equations.

I'm used to solving tridiagonal sets of equations for a similar application and use the standard thomas algorithm. I've heard there's a similar method for the quad case but I can't seem to find the particular algorithm anywhere.

I realise that I could come up with something from first principles but feel that an efficient, tried and tested method would be better than anything I'd cobble together myself.

If someone knows of a good algorithm to use could you please point me in the right direction?

Many thanks, Frank

Dear Frank,

You could always make use of relaxation procedures/iterative solvers to get the solution in an efficient manner, rather than inverting a sparse matrix.

Regards,

Ganesh

Hi Ganesh, I'm already using an iterative proceudre. The quadridiagonal equations result from solving one vertical column (in 2D cartesian mesh) for an iterative SLOR algorithm. If I nested one iterative procedure within another I guess this would take a significant amount of computer time.

To be honest I'm not even too sure how I would go about solving a quadradiagonal set of equations with an iterative solver. I'm relatively new to the whole CFD thing and until now have relied upon mainly black box solvers. My previous work required the solution of tridiagonal sets of equations within an SLOR scheme, for which I used the Thomas algorithm, and I've been told that quad-equations can be solved using an extension of this technique. This would make my life much easier as I'm familiar with the technique and it'd mean the structure of my new code would be quite similar to previous ones. Cheers, Frank

Dear Frank,

Please take a look at the book by Pozrikidis Pozrikidis (1998), pp. 144-146. I think the title is Theorretical and Computational Fluid Dynamics, where it tals about a Modified Thomas algorithm for sparse banded matrices, tri/quad/penta. Also note that Thomas algorithm is a special form of Gaussian elimination, and if you are not really worried about the computational speed and efficiency any iterative solver(like Jacobi, Gauss-Siedel) etc.. in conjunction with LU decomposition can do the job. Do take a look at some of the literature on implicit schemes in CFD, for eg. the point implicit method employed routinely by Lohner et.al. among others.

Hope this helps

Regards,

Ganesh