Dettagli sull'Insegnamento per l'A.A. 2018/2019

Prerequisiti

Conoscenze di base di Analisi Numerica ed equazioni differenziali.

Obiettivi

Questo è un corso monografico sull'approssimazione numerica di equazioni differenziali che intende fornire sia aspetti di base sulla tematica, sia aspetti più avanzati. Unitamente allo studio teorico dei metodi numerici e delle loro proprietà, il corso intende anche fornire gli strumenti di base per il disegno di software matematico che implementi i metodi studiati e l'analisi dei risultati su una selezione di problemi test rilevanti.

Sillabo

• The problem: Hadamard well-posed differential problems. Picard-Lindelof iterations. Existence and uniqueness of solutions. Continuous dependence on initial data and vector fields. Gronwall lemma. Examples of differential problems of interest in real applications.
• Basics on discretization of differential problems: difference equations, consistency, stability, convergence.
• Runge-Kutta methods. Formulation of methods. Butcher theory of order conditions, rooted trees and B-series. Error estimations, Richardson extrapolation, embedded methods. Stepsize control. Collocation methods. Implicit Runge-Kutta methods. Stability analysis.
• Linear multistep methods. Adams-Bashforth methods. Analysis of convergence, Lax theorem. Variable stepsize, Nordsieck formulation. Analysis of stability, boundary locus. BDF methods.
• Stiff problems. A-stability, L-stability, stiffly-stable methods. Schur polynomials, Routh-Hourwitz criterion.
• Structure-preserving approximation: the case of Hamiltonian problems. Basics on Hamiltonian dynamics, invariant preservation. Symplectic numerical methods. The non-symplecticity of linear multistep methods. Long-term properties of symplectic integrators. Quasi-conservation of invariants. Dynamics of multistep methods. Underlying one-step method. Backward error analysis.
• Numerical approximation of stochastic differential equations. Strong and weak solutions. Euler-Maruyama method, Milstein method. Mean-square convergence. Linear multistep stochastic methods. Stochastic Runge-Kutta methods. Mean-square and asymptotic stability.
• Oscillatory differential equations. Modulated Fourier series expansions. Long-term near conservation of total and oscillatory energy.

Descrittori di Dublino

Alla fine del corso, lo studente dovrebbe

Testi di riferimento

• E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations , Springer. 2006.    
• E. Hairer, S.P. Norsett, G. Wanner, Solving ordinary differential equations I. Nonstiff problems , Springer. 1993.    
• E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems , Springer. 1996.    

Modalità d'esame

Esame orale e un progetto finale che consiste nell'applicazione del software sviluppato ad alcuni problemi test scelti. Il progetto verrà discusso nello stesso giorno dell'esame orale.

Aggiornamenti alla pagina del corso

• A.A. 2017/2018
• A.A. 2018/2019
• A.A. 2019/2020