TITLE: Newton-Krylov continuation of periodic orbits for Navier-Stokes flows AUTHORS: J. S\'anchez^(1), M. Net^(1), B. Garc{\'\i}a-Archilla^(2), C. Sim\'o^(3) (1) Departament de F\'{\i}sica Aplicada, Universitat Polit\`ecnica de Catalunya, Jordi Girona, 1--3, m\`odul B4--B5, Campus Nord, 08034 Barcelona (Spain) (2) Departamento de Matem\'atica Aplicada II, Universidad de Sevilla, Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092 Sevilla (Spain) (3) Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08071 Barcelona (Spain) ABSTRACT: Efficient numerical algorithms for the continuation of periodic orbits of high- dimensional dissipative dynamical systems, and for analyzing their stability are presented. They are based on shooting, Newton-Krylov and Arnoldi methods. A thermal convection fluid dynamics problem, which has a rich bifurcation diagram due to symmetries, has been used as test. After a pseudo-spectral discretization of the equations a system of dimension $O(10^4)$ has been obtained. The efficiency of the algorithms, which allows the unfolding of a complex diagram of periodic orbits, makes the methods suitable for the study of large nonlinear dissipative partial differential equations.