Upwind iteration methods for the cell vertex scheme in one dimension

Abstract

This paper describes and analyses a series of methods for solving the algebraic equations obtained from the cell vertex finite volume discretisation in one dimension. The objective is to explore the possibilities for improved iteration methods that may be applied to cell vertex discretisations of the Navier-Stokes equations in higher dimensions. In general there is no natural one-to-one correspondence between cellbased residuals and nodal unknowns for this system. In order to devise iteration schemes it is therefore necessary to provide a mapping between cells and nodes. The family of methods introduced here is based on the application of standard iterative techniques to a nodal residual formed of a combination of neighbouring cell-based residuals. It includes the familiar Lax-Wendroff iteration, upwind iteration schemes, and marching schemes capable of attaining convergence rates independent of the number of algebraic equations. The aim in each case is to set to zero the residual for each cell, apart from exceptional cells such as those containing shocks. The final results show that matrix-based upwind iteration methods, using cell residuals modified to take account of critical points and applying several local iterations, converge in around 15 iterations.