Comparison of Newton's and quasi-Newton's method solvers for the Navier-Stokes equations

Abstract

The results of a computational evaluation of several Newton's and quasi-Newton's method solvers are discussed and analyzed. Computer time and memory requirements for iterating a solution to the steady state are recorded for each method. Roe's flux-difference splitting together with the Spekreijse/Van Albada continuous limiter is used for the spatial discretization. Sparse matrix inversions are performed using a modified version of the Boeing real sparse library routines and the conjugate gradient squared algorithm. The methods are applied to exact and approximate Newton's method Jacobian systems for flat plate and flat-plate/wedge-type ge- ometries. Results indicate that the quasi-Newton's method solvers do not exhibit quadratic convergence, but can be more efficient than the exact Newton's method in select cases. CONSISTENT goal of computational fluid dynamics (CFD) research has been to improve the efficiency of numerical algorithms. One measure of this efficiency is the log of the residual, which varies linearly with iteration count for typical CFD techniques. These methods can be expected to converge at the same rate no matter how close they are to the final solution. An exception to this behavior is the quadratic convergence Newton's method for which the convergence rate actually increases as the final solution is approached. When quadratic convergence has been attained, the number of cor- rect digits in the solution doubles with each iteration. This behavior has been illustrated by several researchers who have obtained solutions of the potential equation,1 the Euler equa- tions,2'5 and the Navier-Stokes equations.2'4'6'13 Unfortu- nately, the number of iterations is a less-important measure of the efficiency of a code than the total computation time. Quadratic convergence rates are typically accompanied by significant increases in algorithm complexity and hence in- creased per-iteration computational requirements. It was the intent of this research to explore how the conver- gence rate and per-iteration efficiency vary for similar New- ton's and quasi-Newton's method solvers so that the most efficient computational strategy can be identified. To accom- plish this goal, the quadratic convergence Newton's method solver developed by Orkwis and McRae68 was modified to create several quasi-Newton's method solvers. Approxima- tions were made to the form of the Jacobian and to the Jacobian matrix inversion process. The resulting schemes are evaluated based on their robustness and efficiency for two test problems.