Nks Methods on Unstructured

We review and extend to the compressible regime an earlier parallelization of an implicit incompressible unstructured Euler code KKS98], and solve for ow over an M6 wing in subsonic, transonic, and supersonic regimes. While the parallelization philosophy of the compressible case is identical to the incompressible, we focus here on the nonlinear and linear convergence rates, which vary in diierent physical regimes, and on comparing the performance of currently important computational platforms. Multiple-scale problems should be marched out at desired accuracy limits, and not held hostage to often more stringent explicit stability limits. In the context of inviscid aerodynamics, this means evolving transient computations on the scale of the convective transit time, rather than the acoustic transit time, or solving steady-state problems with local CFL numbers approaching innnity. Whether time-accurate or steady, we employ Newton's method on each (pseudo-)timestep. The coupling of analysis with design in aerodynamic practice is another motivation for implicitness. Design processes that make use of sensitivity derivatives and the Hessian matrix require operations with the Jacobian matrix of the state constraints (i.e., of the governing PDE system); if the Jacobian is available for design, it may be employed with advantage in a nonlinearly implicit analysis, as well. Implicit methods tend to contain global operations, which makes them challenging to parallelize. Nevertheless, the increasing resolution requirements of PDE analyses require access to the large memories provided by parallelism. Our framework for an implicit PDE solution algorithm, with pseudo-timestepping to advance towards an assumed steady state, has the form: (1 t `)u ` +f(u `) = (1 t `)u `?1 ; where t ` ! 1 as`! 1; where u represents the fully coupled vector of unknowns, and the steady-state solution satisses f (u) = 0. Each member of the sequence of nonlinear problems, ` = 1; 2; : : :, is solved with an inexact Newton method. The resulting Jacobian systems for the Newton corrections are solved with a Krylov method, relying directly only on matrix-free operations. The Krylov method needs to be preconditioned for acceptable inner iteration convergence rates, and the preconditioning can be the \make-or-break" feature of an implicit code. A good preconditioner saves time and space by permitting fewer iterations in the Krylov loop and smaller storage for the Krylov subspace. An additive Schwarz preconditioner Cai89] accomplishes this in a concurrent, localized manner, with an approximate solve in each subdomain of a partitioning of …