Initial-value problems: Runge-Kutta, extrapolation and multistep methods. Stable methods for stiff problems. Boundary-value problems: Shooting and multiple shooting. Difference schemes, collocation. Analysis. Conditioning of boundary value problems. Consistency, stability and convergence for both initial and boundary value problems. Fourier transform tecniques. Fourier analysis, Fourier spectral methods. Geometric integrators. Lie group methods, symplectic methods, Magnus series method.