NEWTON–GMRES ALGORITHM APPLIED TO COMPRESSIBLE FLOWS

SUMMARY This paper addresses the resolution of non-linear problems arising from an implicit time discretization in CFD problems. We study the convergence of the Newton-GMRES algorithm with a Jacobian approximated by a finite difference scheme and with restarting in GMRES. In our numerical experiments we observe, as predicted by the theory, the impact of the matrix-free approximations. A second-order scheme clearly improves the convergence in the Newton process. Many scientific applications lead to a non-linear system of equations. We consider here the numerical simulation of steady state compressible flows. Implicit time discretizations allow us to use large time steps. On the other hand, at each time step a non-linear system of equations must be solved. Because of memory requirements, we want to use a so-called matrix-free algorithm. Several authors (see e.g. References 1 and 2) have considered inexact Newton methods where the Newton equations are solved approximately by an iterative solver. Moreover, since the Jacobian is required only through a matrixvector product, it can be approximated by a finite difference scheme.334 The resulting matrix-free algorithm, which we call Newton-MF-GMRES, has been studied there with no restarting in GMRES. Here we extend these results to GMRES with restarting, denoted GMRES(m), as designed in Reference 5. Global convergence of Newton can be enhanced by a line search backtracking procedure provided that the approximate solution given by the iterative solver is a descent direction.6 We give a sufficient condtion on the stopping criterion of GMRES(m) to guarantee this result. The quadratic local convergence of the basic Newton iterations is no longer achieved with the Newton-MF-GMREiS method. As in Reference 3, but in the context of restarting, we give here sufficient conditions on the stopping criterion and the approximation of the Jacobian to obtain a linear local convergence. We introduce a centred second-order difference quotient to approximate the Jacobian. This scheme is more expensive than the usual first- order difference quotient, but it is more accurate and leads to a better Newton convergence. We apply the Newton-MF-GMRES(m) algorithm to the numerical solution of the compressible Navier-Stokes equations. We present results for two steady state problems. We study in detail the convergence of Newton and GMRES for one implicit time step and also for the stationary non-linear