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The aim of this study is to implement Newton-GMRES method into supersonic flows as an alternative to Newton method which is a reliable and efficient solver. The flow analysis is based on the three dimensional Euler equations. The objective of this study is compare the performances of Newton and Newton-GMRES methods in a 3-D nozzle design. The Jacobian matrix needed for Newton method is evaluated analytically. Newton-GMRES method is a matrix free solution technique hence it does not require the Jacobian matrix evaluation. A gradient based method is used for numerical optimization. Sensitivities are evaluated by using adjoint method. Design optimization is conducted by Newton's method. I. IntroductionRecent advances in computer technology and solution algorithms allow efficient solution of very large linear systems of equations. These advances have been motivating researchers to develop implicit algorithms to solve the flow equations since usage of implicit methods is more beneficial compared to the explicit ones. Implicit flow solvers are more stable and the residual can be reduced to very low values within a small number of iterations. The equations of different disciplines can be strongly coupled with flow equations in an implicit algorithm.Characterized by high Mach number and high enthalpy, hypersonic/supersonic flows are special flow types in which physic-chemical and numerical modeling is difficult. Larger numbers of species and reaction equations are needed to model hypersonic flows. In such flow conditions, due to the numerical stiffness of the system, convergences problems may be observed. Implicit methods may have more advantages to avoid these problems. Providing quadratic convergence, Newton method is one of the preferred algorithms to solve non-linear equations. The requirement for the evaluation and the solution of large Jacobian matrix is the main disadvantage of Newton method. The Jacobian matrix is evaluated by taking the derivatives of residual vector with respect to flow variable vector. Although the size of this matrix can be very large, it is sparse in the most of the flow problems. The selection of good initial solution is important in Newton method. If the initial solution is not chosen properly, Newton method may diverge. To keep the advantages and to avoid disadvantages of Newton method, in recent years, Jacobian-free Newton methods are getting more attention. One of these methods is Newton-GMRES method. Newton-GMRES method has been developed and applied in different areas of CFD. Some of these physics models were given in Knoll and Keyes 1 . A Newton-Krylov method can be summarized as an iterative method which is a combination of a linear iterative method. GMRES is used for solving linear systems which is a Krylov subspace method, found by Saad and Schultz 2 .In this study, the performances of Newton and Newton-GMRES methods are compared for the flow analysis and design optimization of three dimensional nozzles. For flow analysis, a cell centered finite volume code is developed by using t...

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