WELL POSEDNESS OF THE PROBLEM
Hadamard well-posed differential problems. Picard-Lindelof iterations, analysis of their convergence, long-term analysis. Existence and uniqueness of solutions. Continuous dependence on initial data and vector fields. Gronwall lemma. Examples of differential problems of interest in real applications.

DIFFERENCE EQUATIONS
Linear difference equations. Space of solutions and its basis. Casorati matrix. General solution of homogeneous and inhomogeneous equations. Characteristic equations.

ONE-STEP DISCRETIZATIONS
Discretization of the domain, numerical grids. Euler method. Consistency, zero-stability and convergence. Local truncation error. Trapezoidal method. Fixed point iterations for implicit schemes.

LINEAR MULTISTEP METHODS
Multistep discretizations, starting procedure. First and second characteristic polynomials. Implicit methods and convergence analysis of fixed point iterations. Local truncation error. Linear difference operator. Consistency. Zero-stability and root condition. Convergence and order of convergence. Lax equivalence theorem. Methods arising from quadrature formulae. Adams-Bashforth and Adams-Moulton schemes. Dahlquist barriers.

RUNGE-KUTTA METHODS
Definition and classification of Runge-Kutta methods. Order conditions. Butcher theory of order: B-trees, square bracket representation, order, symmetry and density; elementary differentials; elementary weights. B-series. Convergence analysis. Butcher barriers. Implicit methods: Gauss, Radau and Lobatto methods.

LINEAR STABILITY ANALYSIS AND STIFF PROBLEMS
Dahlquist test problem. Linear stability analysis for linear multistep and Runge-Kutta methods. Regions and intervals of absolute stability. Stepsize restrictions. Boundary locus. A-stability, L-stability, A(alpha)-stability. Rational approximations of the exponential, Padé approximations. Stiff problems, stiffness ratio. A-priori Prothero-Robinson analysis.

VARIABLE STEPSIZE IMPLEMENTATION
Predictor-corrector schemes. Error estimates. Stepsize control strategy: computation of the optimal stepsize, PI and PID controller. Newton iterations for implicit Runge-Kutta methods.

GEOMETRIC NUMERICAL INTEGRATION
Numerical conservation of contractivity for dissipative problems. One-sided Lipschitz. Logarithmic norm. G-stability, B-stability, algebraic stability. Differential problems with first integrals. Conservation of linear invariants. Conservation of quadratic invariants. Symplectic Runge-Kutta methods. Hamiltonian problems: symplecticity of the flow, preservation of the Hamiltonian. Long-time behavior of symplectic methods to Hamiltonian problems. Modified differential equations. Benettin-Giorgilli theorem.