Introduce to the study and implementation of more advanced numerical approximation techniques, in particular related to approximate solution of ordinary differential equations, and to a further advanced topic to be chosen between the optimization and the fundamentals of approximation of partial differential equations.
CACACE SIMONEteacher profile teaching materials
Mutuazione: 20410420 AN420 - ANALISI NUMERICA 2 in Scienze Computazionali LM-40 CACACE SIMONE
ProgrammeOrdinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")
Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)
Core DocumentationRoberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf
Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf
Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html
Additional notes given by the teacher
Type of delivery of the coursefrontal teaching course
Type of evaluationtheoretical oral test and evaluation of a matlab project