Fully coupled finite volume solutions of the incompressible Navier–Stokes and energy equations using an inexact Newton method

Abstract

AbstractAn inexact Newton method is used to solve the steady, incompressible Navier–Stokes and energy equation. 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.