"Convergence of iterative methods for the solution of the steady quasi-one-dimensional nozzle problem with shocks is considered. The finite-difference algorithms obtained from implicit schemes are used to approximate both the Euler and Navier-Stokes Equations. These algorithms are investigated for stability and convergence characteristics. The numerical methods are broken down into their matrix-vector components and then analyzed by examining a subset of the eigensystem using a method based on the Arnoldi process. The eigenvalues obtained by this method are accurate to within 5 digits for the largest ones and to within 2 digits for the ones smaller in magnitude compared the elgenvalues obtained using the full Jacobian. In the analysis we examine the functional relationship between the numerical parameters and the rate of convergence of the iterative scheme. Acceleration techniques for iterative methods like Wynn's e-algorithm are also applied to these systems of difference equations in order to accelerate their convergence. This acceleration translates into savings in the total number of iterations and thus the total amount of computer time required to obtain a converged solution. The rate of convergence of the accelerated system is found to agree with the prediction based on the eigenvalues of the original iteration matrix. The ultimate goal of this study is to extend this elgenvalue analysis to multi-dimensional problems and to quantitatively estimate the effects of different parameters on the rate of convergence."
Mohammad Saleem, A Cheer, M Hafez, and T Pulliam. "Analysis of the convergence history of fluid flow through nozzles with shocks" AIAA Meeting Papers, 1st National Fluid Dynamics Conference (1988): 620-628. doi:10.2514/6.1988-3795