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.
teacher profile teaching materials
Finite difference methods for Ordinary Differential Equations: Euler method. Consistency, stability, absolute stability. Second-order Runge-Kutta methods. Implicit one-step methods: backward Euler, Crank-Nicolson. Convergence of one-step methods.
Multistep methods: general structure, complexity, absolute stability. Stability and consistency for multistep methods. Adams methods. BDF methods. Predictor-Corrector methods. (reference: chapter 7 of the notes "Appunti del corso di Analisi Numerica")
Difference schemes for Partial Differential Equations
General concepts about finite difference approximations. Semi-discrete approximations and their convergence. Lax-Richtmeyer theorem.
The advection equation: method of characteristics. Semi-dicrete and fully discrete upwind method, consistency and stability. The heat equation: Fourier approximation. Centred finite difference approximation, consistency and stability. Poisson equation: Fourier and centred difference approximations, convergence study. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected material from chapters 1, 2, 3, 12, 13)
N.B.: References are provided with respect to the course notes.
Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf
Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html
Additional notes provided by the teacher
Mutuazione: 20402092 AN420 - ANALISI NUMERICA 2 in Scienze Computazionali LM-40 FERRETTI ROBERTO, CACACE SIMONE
Programme
Ordinary Differential EquationsFinite difference methods for Ordinary Differential Equations: Euler method. Consistency, stability, absolute stability. Second-order Runge-Kutta methods. Implicit one-step methods: backward Euler, Crank-Nicolson. Convergence of one-step methods.
Multistep methods: general structure, complexity, absolute stability. Stability and consistency for multistep methods. Adams methods. BDF methods. Predictor-Corrector methods. (reference: chapter 7 of the notes "Appunti del corso di Analisi Numerica")
Difference schemes for Partial Differential Equations
General concepts about finite difference approximations. Semi-discrete approximations and their convergence. Lax-Richtmeyer theorem.
The advection equation: method of characteristics. Semi-dicrete and fully discrete upwind method, consistency and stability. The heat equation: Fourier approximation. Centred finite difference approximation, consistency and stability. Poisson equation: Fourier and centred difference approximations, convergence study. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected material from chapters 1, 2, 3, 12, 13)
N.B.: References are provided with respect to the course notes.
Core Documentation
Roberto Ferretti, "Appunti del corso di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdfRoberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf
Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html
Additional notes provided by the teacher
Type of evaluation
The theoretical part consists in a written 2h30 examination; a collection of the exam themes can be found on the course page. The laboratory part requires an additional test, consisting in a short (2h) Matlab coding exam, on a topic close to what has been treated in the lab classes, but possibly with slight modifications. teacher profile teaching materials
Mutuazione: 20402092 AN420 - ANALISI NUMERICA 2 in Scienze Computazionali LM-40 FERRETTI ROBERTO, CACACE SIMONE