Quasi-Newton Iterations, preconditioning, and numerical diffusion for a 4th-order operator compact compressible Navier-Stokes solver

This thesis deals with improvements to a Compressible Navier Stokes (N-S) solver. These code modifications will result in time performance and accuracy gain. The improvements are geared towards performance in the incompressible regime. Typical compressible N-S solvers converge very slowly in incompressible flow regimes. This leads to long run times for compressible problems with embedded low speed flows such as near stagnation points and boundary layers. In fact, separate incompressible codes are written for low speed problems which have faster convergence properties. The goal of the modifications is to provide a single N-S code which will not only have improved convergence in incompressible flow regimes, but also could be used for compressible problems without having to resort to a separate code. A second goal is to improve the accuracy of the computations for both steady and unsteady problems. Recently, two techniques have been reported which hold the promise of significant improvements in both convergence and accuracy: artificial time stepping (dual time stepping, inner iterations, and other schemes of the same sort) and preconditioning. The first deals with the splitting error of approximate factorization algorithms, and the second alleviates the problem of the slow convergence of the low incompressible regime. In this thesis, three areas are explored. First, Quasi-Newton Iterations (a form of artificial time stepping) are applied to a vectorized N-S solver which uses a 4th-order compact discretization. This modification is designed to remove the Approximate Factorization (AF) splitting error which will also speed up convergence by allowing a larger time step. In the second area of exploration, a new modification to Merkle's preconditioning matrix is implemented. The p coefficient is modified to take into account the curvilinear equations and the artificial implicit numerical diffusion terms used in the discretization. This new preconditioning was named the Psi revision. In the third area of effort, a proper numerical diffusion of the Operator Compact of the 4th-order (OC4) is derived. It shows that the assumption currently in place, using a similar numerical diffusion for the OC4 as for the 2nd-order central scheme, is correct.