Preconditioning for the Fluid Dynamic Equations

The selection of articles for this issue of the ICASE Research Quarterly departs from the previous policy of presenting technical articles covering a variety of active research areas at ICASE; instead, the articles are devoted to a single topic: convergence acceleration in CFD. The articles for this special issue, solicited by Dr. Jim Jones, Staa Scientist at ICASE, discuss convergence acceleration through matrix precondi-tioning and improved multigrid methods. The convergence rate achieved in practical calculations is often much slower than the theoretical rate. The articles in this issue identify some causes of slow convergence and ooer suggestions which promise considerable improvement. We welcome our readers' reactions to this type of issue and their suggested topics for possible future special issues. Professor Bokhari has also been elected Fellow of the IEEE for his contribution to the mapping problem in parallel and distributed computing. Professor Bokhari began this work at ICASE in 1978. His 1979 ICASE report on the mapping problem was published in the IEEE Transactions on Computers , and became one of the most frequently cited papers on this problem. Scientists aaliated with ICASE who also contributed to this work include current ICASE staa members Introduction Over the past years, several preconditioning methods have appeared which attempt to solve nearly incompressible ow problems with numerical algorithms designed for compressible ows. These developments are motivated by several observations. First, in some problems, part of the ow is incompressible with locally low Mach numbers , but signiicant compressibility eeects occur elsewhere in the ow. A typical example in aerodynamics is the ow over a wing at a high angle of attack. Other examples include duct ow, in which most of the ow is highly subsonic, but sharp changes in the duct geometry cause local regions of supersonic ow. Low speed ows may be compress-ible because of surface heat transfer or volumet-ric heat addition. Finally, engineers prefer to use existing compressible ow codes over the broadest possible range of ow conditions in order to avoid using multiple ow codes. The diiculty in solving the compressible equations for low Mach numbers originates with the large disparity between the acoustic wave speed u + a and the convective wave speed u. Precon-ditioning changes the eigenvalues of the system of compressible ow equations to mitigate this disparity. The time derivatives are premultiplied by a matrix which reduces the acoustic wave speed towards the uid speed. We …