Comparison of standard and matrix-free implementations of several Newton-Krylov solvers

Abstract

Fully coupled Newton's method is combined with conjugate gradient-like iterative algorithms to form inexact Newton-Krylov algorithms for solving 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 correction in an enclosed cavity is solved. Three conjugate gradient-like algorithms are selected from a clans of algorithms based upon the Lanczos biorthogonalization procedure; namely, the conjugate gradients squared algorithm, the transpose-free quasi-minimal-residual algorithm, and a more smoothly convergent version of the biconjugate gradients algorithm. A fourth algorithm is based upon the Arnoldi process, namely the popular generalized minimal residual algorithm (GMRES). The performance of a standard inexact Newton's method implementation is compared with a matrix-free implementation. Results indicate that the performance of the matrix-free implementation is strongly dependent upon grid size (number of unknowns) and the selection of the conjugate gradient-like method. GMRES appeared to be superior to the Lanczos based algorithms within the context of a matrix-free implementation