FULLY COUPLED FINITE-VOLUME SOLUTIONS OF THE INCOMPRESSIBLE NAVIER-STOKES AND ENERGY EQUATIONS USING AN INEXACT NEWTON METHOD

An inexact Newton method is used to solve the steady, incompressible Navier-Stokes and energy equations. Finite volume differencing is employed on a staggered grid using the power law scheme of Patankar. Natural convection in an enclosed cavity is studied as the model problem. Two conjugate-gradient-like algorithms based upon the Lanczos biorthogonalization procedure are used to solve the linear systems arising on each Newton iteration, The first conjugate-gradient-like algorithm is the transpose-free quasi-minimal residual algorithm (TFQMR) and the second is the conjugate gradients squared algorithm (CGS). Incomplete lower-upper (ILU) factorization of the Jacobian matrix is used as a right preconditioner. The performance of the Newton-TFQMR algorithm is studied with regard to different choices for the TFQMR convergence criteria and the amount of fill-in allowed in the ILU factorization. Performance data are compared with results using the Newton-CGS algorithm and previous results using LINPACK banded Gaussian elimination (direct-Newton). The inexact Newton algorithms were found to be CPU competetive with the direct-Newton algorithm for the model problem considered. Among the inexact Newton algorithms, Newton-CGS outperformed Newton-TFQMR with regard to CPU time but was less robust because of the sometimes erratic CGS convergence behaviour.