UPWIND ITERATION METHODS FOR THE CELL VERTEX SCHEME IN ONE-DIMENSION

This paper describes and analyses a series of methods for solving the algebraic equations obtained from the cell vertex finite volume discre tisation in one dimension. The objective is to explore the possibiliti es for improved iteration methods that may be applied to cell vertex d iscretisations of the Navier-Stokes equations in higher dimensions. In general there is no natural one-to-one correspondence between cell-ba sed residuals and nodal unknowns for this system. In order to devise i teration schemes it is therefore necessary to provide a mapping betwee n cells and nodes. The family of methods introduced here is based on t he application of standard iterative techniques to a nodal residual fo rmed of a combination of neighbouring cell-based residuals. It include s the familiar Lax-Wendroff iteration, upwind iteration schemes, and m arching schemes capable of attaining convergence rates independent of the number of algebraic equations. The aim in each case is to set to z ero the residual for each cell, apart from exceptional cells such as t hose containing shocks. The final results show that matrix-based upwin d iteration methods, using cell residuals modified to take account of critical points and applying several local iterations, converge in aro und 15 iterations. (C) 1994 Academic Press, Inc.