Inexact Newton's method solutions to the incompressible Navier-Stokes and energy equations using standard and matrix-free implementations

Fully implicit Ncwton's mcthod is coupled with conjugate gradient-likc itcrativc algorithms to form inexact Newton algorithms for solving the stcady, incompressible, Navicr-Stokes and encrgy equations in primitive variables. Finitc volumc diffcrcncing is employed using the power law convection-diffusion scheme on a uniform, but slaggcrcd mesh. The wcll known model problcm of natural convection in an enclosed cavity is solvcd. Thrce conjugatc gradient-like algorithms arc sclcctcd from a class of algorithms bascd upon thc Lanczos biorthogonalization proccdurc; namely, the conjugate gradicnts squarcd algorithm (CGS), the transpose-frce quasi-minimal rcsidual algorithm (QMRCGS), and a morc smoothly convcrgcnt version of thc bi-conjugate gradicnts algorithm (Bi-CGSTAB). A fourth algorithm is bascd upon the Arnoldi proccss, namcly the popular gcncralized minimal rcsidual algorithm (GMRES). The performance of a standard incxact Ncwton's mcthod implementation is comparcd with a matrix-frec implementation. Rcsults indicatc that thc performance of the matrix-free implemcntation is strongly dcpcndcnt upon grid size (numbcr of unknowns) and thc sclcction of the conjugatc gradient-like mcthod. GMRES was found to be supcrior to the Lanczos bascd algorithms within the contcxt of a matrix-frcc implcmcnlation. Among the Lanczos bascd algorithms, QMRCGS and Bi-CGSTAB wcrc bctter suitcd to thc matrix-frce implcmcntation than CGS bccausc of thcir smoothcr convcrgcncc behavior.

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