A generalized Conjugate Gradient like method is used to solve the linear systems
of equations formed at each time-integration step of the unsteady, two-dimensional,
compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are
cast in an implicit, upwind finite-volume, flux split formulation. Preconditioning
techniques are employed with the Conjugate Gradient like method to enhance the
stability and convergence rate of the overall iterative method. The superiority
of the new solver is established by comparisons with a conventional Line GaussSeidel
Relaxation (LGSR) solver. Comparisons are based on 'number of iterations
required to converge to a steady-state solution' and 'total CPU time required for
convergence'. Three test cases representing widely varying flow physics are chosen
to investigate the performance of the solvers. Computational test results for very
low speed (incompressible flow over a backward facing step at Mach 0.1), transonic
flow (trailing edge flow in a transonic turbine cascade) and hypersonic flow (shockon-
shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For
the 1vfach 0.1 case, speed-up factors of 30 (in terms of iterations) and 20 (in terms
of CPU time) are found in favor of the new solver when compared with the LGSR
solver. The corresponding speed-up factors for the transonic flow case are 20 and
18, respectively. The hypersonic case shows relatively lower speed-up factors of 5
and 4, respectively. This study reveals that preconditioning can greatly enhance
the range of applicability and improve the performance of Conjugate Gradient like
methods.
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