COMPARISON OF STANDARD AND MATRIX-FREE IMPLEMENTATIONS OF SEVERAL NEWTON-KRYLOV SOLVERS

Fully coupled Newton's method is combined with conjugate gradient-like iterative algorithms to form inexact Newton-Krylov algorithms for sol ving the steady, incompressible, Navier-Stokes and energy equations in primitive variables. Finite volume differencing is employed using the power law convection-diffusion scheme on a uniform but staggered mesh . The well-known model problem of natural convection in an enclosed ca vity is solved. Three conjugate gradient-like algorithms are selected from a class of algorithms based upon the Lanczos biorthogonalization procedure; namely, the conjugate gradients squared algorithm, the tran spose-free quasi-minimal-residual algorithm, and a more smoothly conve rgent version of the biconjugate gradients algorithm. A fourth algorit hm is based upon the Arnoldi process, namely the popular generalized m inimal residual algorithm (GMRES). The performance of a standard inexa ct Newton's method implementation is compared with a matrix-free imple mentation. Results indicate that the performance of the matrix-free im plementation is strongly dependent upon grid size (number of unknowns) and the selection of the conjugate gradient-like method. GMRES appear ed to be superior to the Lanczos based algorithms within the context o f a matrix-free implementation.