Relaxation methods for unfactored implicit upwind schemes

Abstract

Relaxation methods are presented for unfactored implicit upwind schemes for hyperbolic equations. The theoretical bases are explained using linear and nonlinear scalar equations; construction of the method for the unsteady Euler equations (nonlinear system) is but a natural extension. One of the important advantages of the above methods vis a vis factored implicit schemes is the possibility of faster convergence to steady state, as illustrated by the results. Several classes of relaxation schemes such as pointwise, linewise, Gauss-Seidel, and non-Gauss-Seidel methods are discussed, along with various strategies for convergence.