Ordering and incomplete factorization issues for matrices arising from the TRANAIR CFD code. A locally reened rectangular grid nite element method: Application to computational uid dynamics and computational physics. A comparison of domain decomposition techniques for elliptic partial diierential equations and their parallel implementation. Inexact Newton's method solutions to the incompressible Navier-Stokes and energy equations using standard and matrix-free implementations. 16 eration counts and/or memory consumption. However, the plethora of parameters can be exploited, in principle, to produce optimal tradeoos in space and time for a given problem class. Though parametric tuning is important to performance, conservative robust choices are not diicult. methods for the unsteady compressible Navier-Stokes equations on unstructured meshes. A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems. 15 Table 8: Wall-clock performance and relative parallel eeciency for unstructured Euler code on an Intel Paragon. We have shown that steady aerodynamics problems in two diierent formulations (full potential and Euler) can be eeectively solved, and cost-eeectively solved in parallel, by NKS methods. The NK technique has been compared with V-cycle multigrid on Euler and Navier-Stokes problems without parallelizing the preconditioning in 14, 20]. For a subsonic unstructured grid example, NK trails multigrid in execution time by a factor of only about 1.5. This penalty can be accepted when it is realized that the NK method has the advantage of doing all of its computation without generation of a family of coarse unstructured grids (which is diicult for three-dimensional unstructured grids). This work has been extended to three-dimensional problems in 20]. Large-scale time-dependent problems suuering from multiple scales often require parallel implicit algorithms. The KS technique has been shown eeective in the unsteady Navier-Stokes context in 3]. In 3], two of the same parameters explored herein (level of ll in the local ILU factorizations and subdomain overlap) are varied to produce a Schwarz precondi-tioner whose strength can be adjusted to adapt to the varying time-evolving ill-conditioning of the linear system arising at each implicit time step. A variety of CFD applications are (or have inner) nonlinear elliptically-dominated problems amenable to solution by NKS algorithms, which are characterized by low storage requirements (for an implicit method) and locally concentrated data dependencies with small overlaps between the preconditioner blocks. The addition of a global coarse grid in the Schwarz preconditioner is often eeective, where architecturally convenient. A deterrent to the widespread adoption of NKS …

In this paper the use of a new out-of-core sparse matrix package for the numerical solution of partial differential equations involving complex geometries arising from aerospace applications is discussed. The sparse matrix solver accepts contributions to the matrix elements in random order and assembles the matrix using fast sort/merge routines. Fill-in is reduced through the use of a physically based nested dissection ordering. For very large problems a drop tolerance is used during the matrix decomposition phase. The resulting incomplete factorization is an effective preconditioner for Krylov subspace methods, such as GMRES. Problems involving 200,000 unknowns routinely are solved on the Cray X-MP using 64MW of solid-state storage device (SSD).

Attention is given to a new approach to solving full potential equations about arbitrary configurations. Numerical algorithms from such fields as finite elements, preconditioned Krylov subspace methods, discrete Fourier analysis, and integral equations are combined to take advantage of the size and speed of current and emerging supercomputers. On the basis of this appraoch, a robust, efficient and easy to use computer code referred to as TRANAIR has been developed for transonic analysis of complex geometries.