This course aims to provide an advanced knowledge regarding the numerical solution of initial value problems for ordinary differential equations. D1 - Knowledge and understanding skills By the end of the course, the student will have to know and be familiar with the theory of Runge-Kutta methods with particular emphasis on the computation of the order conditions, on the construction of suitable explicit and implicit methods and on the stability properties. He/she will also have to know the multi-step methods at a basic level. D2 - Applying knowledge and understanding The student will have to be able to face and solve exercises, questions and problems related to the topics treated in the course. D3 - Making judgements Among the studied methods, the student will have to be able to select suitably the ones that better approximate the solutions of ordinary differential equations which have got some possible particular characteristics. D4 - Communication skills The student will have to be able to describe the acquired techniques and the issues faced in the course with an appropriate vocabulary. D5 - Learning skills The student will have to be able to read and understand books and articles which treat the topics learnt in the course and, also, to be able to look further into them. He/she will have to be able to propose on his/her own, possibly with suitable adjustments, the use of the acquired tools for the numerical solution of differential mathematical models which may be met in applied subjects.

Good knowledge of Mathematical Analysis, typically supplied by a "Corso di Laurea triennale" in Mathematics or in Physics or in Engineering. Basic knowledge of Numerical Analysis including the numerical solution of ordinary differential equations

Review on ODEs. - Initial Value Problems and Boundary Value Problems. - Systems of ODEs. - Linear systems of ODEs: the matrix exponential, the Magnus expansion. - Higher order ODEs. - Autonomous ODEs. - Existence and uniqueness results for IVPs: the local scenario, the global scenario, the Picard-Lindelof iteration. - Numerical methods. First methods for ODEs. - The Euler method. - One-step methods: the exact flow map and the approximate flow map, the Assumption of Definiteness, the incremment function, global error and local error, order of a method. - Convergence of one-step methods: convergence of the explicit Euler method, convergence of the implicit Euler method. - The one-leg methods: convergence of one-leg methods. - The theta-methods: convergence of theta methods. Runge-Kutta methods. - Compact form of the RK equations. - The assumption of Definiteness. - Convergence analysis of RK methods. - Order of RK methods: order barrier, autonomous equations, the Albrecht approach, the order conditions up to the order four, the Butcher approach. - Explicit RK methods: order barrier, two-stage explicit Rk methods of order two, three-stage explicit RK methods of order three, Four-stage explicit RK methods of order four, more than four stages. -Implicit RK methods: The collocation approach, Collocation RK methods, construction of collocation methods. Adaptive mesh. - A new global error analysis: the new propagated error and the new local error, a new bound for the global error. - Stepsize control. - How to estimate the local errors: embedded pairs of explicit RK methods, Local extrapolation. - Numerical ODE solvers: relative tolerance, MATLAB ODE solvers. Stiff Equations - Analysis of the stiffness: non-expansive flow maps, stiffness with non-expansive flow maps - The complex scalar linear ODE: the stability function of a RK method, the stability region, the stability requirement, the A-stability. - Linear systems of ODEs. Linear Multistep methods - Linear k-step methods. - The Assumption of Definiteness and the Implicit form of the approximate flow map. - Convergence and consistency. - Convergence analysis: zero stability. - Order conditions: order conditions by the polynomials ρ and σ. - Order Barriers. - Adams methods. - BDF methods. - A comparison between LM methods and RK methods.

J.C. Butcher: The Numerical Analysis of Ordinary Differential Equations, Wiley. K. Dekker and J.G. Verwer: Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland E. Hairer, S.P. Norsett, G. Wanner: Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag. E. Hairer, G. Wanner: Solving Ordinary Differential Equations II, Stiff and Differential Algebraic Problems, Springer-Verlag. Lecture notes.

Review on ODEs. - Initial Value Problems and Boundary Value Problems. - Systems of ODEs. - Linear systems of ODEs: the matrix exponential, the Magnus expansion. - Higher order ODEs. - Autonomous ODEs. - Existence and uniqueness results for IVPs: the local scenario, the global scenario, the Picard-Lindelof iteration. - Numerical methods. First methods for ODEs. - The Euler method. - One-step methods: the exact flow map and the approximate flow map, the Assumption of Definiteness, the incremment function, global error and local error, order of a method. - Convergence of one-step methods: convergence of the explicit Euler method, convergence of the implicit Euler method. - The one-leg methods: convergence of one-leg methods. - The theta-methods: convergence of theta methods. Runge-Kutta methods. - Compact form of the RK equations. - The assumption of Definiteness. - Convergence analysis of RK methods. - Order of RK methods: order barrier, autonomous equations, the Albrecht approach, the order conditions up to the order four, the Butcher approach. - Explicit RK methods: order barrier, two-stage explicit Rk methods of order two, three-stage explicit RK methods of order three, Four-stage explicit RK methods of order four, more than four stages. -Implicit RK methods: The collocation approach, Collocation RK methods, construction of collocation methods. Adaptive mesh. - A new global error analysis: the new propagated error and the new local error, a new bound for the global error. - Stepsize control. - How to estimate the local errors: embedded pairs of explicit RK methods, Local extrapolation. - Numerical ODE solvers: relative tolerance, MATLAB ODE solvers. Stiff Equations - Analysis of the stiffness: non-expansive flow maps, stiffness with non-expansive flow maps - The complex scalar linear ODE: the stability function of a RK method, the stability region, the stability requirement, the A-stability. - Linear systems of ODEs. Linear Multistep methods - Linear k-step methods. - The Assumption of Definiteness and the Implicit form of the approximate flow map. - Convergence and consistency. - Convergence analysis: zero stability. - Order conditions: order conditions by the polynomials ρ and σ. - Order Barriers. - Adams methods. - BDF methods. - A comparison between LM methods and RK methods.

Lectures, either of theoretical character or aimed to solve the related exercises.

The teacher distributes the related notes at the end of each topic.

The purpose of the exam, which will be carried on in oral form only, is to evaluate the knowledge of the topics treated in the course and the skills in applying the theory and its methods to the solution of exercises.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs).