an LU approximate factorization (see, for example, Jameson [8]). LU schemes are used in the incompressible meth-A lower-upper/approximate-factorization (LU/AF) scheme is developed for the incompressible Euler or Navier-Stokes equations. ods of [3,5,[9][10][11][12][13]. The treatment of diffusion terms in LU
The LU/AF scheme contains an iteration parameter that can be schemes is considered in [14].
adjusted to improve iterative convergence rate. The LU/AF scheme In the present paper, a modified LU approximate factoris to be used in conjunction with linearized implicit approximations ization is developed for use in conjunction with linearized and artificial compressibility to compute steady solutions, and implicit approximations and artificial compressibility to within sub-iterations to compute unsteady solutions. Formulations based on time linearization with and without sub-iteration and on compute steady or unsteady solutions of the incompress-Newton linearization are developed using spatial difference operaible Navier-Stokes equations. The purpose of the modified tors. The spatial approximation used includes upwind differencing LU/AF scheme is the introduction of an iteration paramebased on Roe's approximate Riemann solver and van Leer's MUSCL ter Ν° which can be adjusted to improve the iterative converscheme, with numerically computed implicit flux linearizations.
gence rate. For one choice of the parameter (Ν° Ο 1), this Simple one-dimensional diffusion and advection/diffusion problems are first studied analytically to provide insight for development scheme is equivalent to symmetric Gauss-Seidel relaxof the Navier-Stokes algorithm. The optimal values of both time ation; for another choice (Ν° Ο 0), the factorization is an LU step and LU/AF parameter are determined for a test problem conanalog of that used in ADI schemes. Optimal convergence sisting of two-dimensional flow past a NACA 0012 airfoil, with a behavior is found to occur at an intermediate value of the highly stretched grid. The optimal parameter provides a consistent parameter. The scheme can be used with or without subimprovement in convergence rate for four test cases having different grids and Reynolds numbers and, also, for an inviscid case. The iteration at each time step. Formulations based on both scheme can be easily extended to three dimensions and adapted time-linearization and Newton linearization are given.