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Scheda Riassuntiva
Anno Accademico 2015/2016
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 096010 - NUMERICAL MODELING OF DIFFERENTIAL PROBLEMS
Docente Formaggia Luca
Cfu 6.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - BV (469) AERONAUTICAL ENGINEERING - INGEGNERIA AERONAUTICA*AZZZZ096010 - NUMERICAL MODELING OF DIFFERENTIAL PROBLEMS

Programma dettagliato e risultati di apprendimento attesi

Numerical Modelling of Differential Problems

Objectives

The course aims at providing the fundamental tools for the numerical simulation of problems governed by partial differential equations. In particular the course addresses elliptic, parabolic and hyperbolic equations and methods for their numerical solution by finite differences, finite volume and finite elements. The issues of consistency, stability and convergence of the methods as well as their practical implementation are analysed in detail.

Description

1)      Short introduction to functional analysis: linear spaces, Hilbert spaces, Sobolev spaces. Concept of internal product and norms. Fundamental inequalities.

2)      Elliptic problem: Laplace equation, convection-diffusion equations.Weak formulation, Galerkin and finite element discretization and the resulting algebraic system. General results of consistency, stability and convergence of the method. Stabilization techniques for convection dominated problems.

3)      Parabolic equations. Weak formulation and finite element discretization. Integration in time. Main convergence results.

4)      Stokes problem. Compatible boundary conditions. Weak formulation. The pressure-velocity coupling: stability condition. Discretization by finite elements. Compatible ad incompatible finite element spaces. Stabilization techniques. The saddle point algebraic problem.

5)      Incompressible Navier-Stokes. The different treatment of the convective term: implicit, semi-explicit, fully explicit. Finite element discretization. Fixed point techniques for the nonlinear term. Semi-Lagrangian schemes. Fractional step methods: Chorin-Temam scheme in its basic and incremental form. Stabilization techniques.

6)      First order linear hyperbolic equations. Boundary conditions. The method of characteristics. Finite volume discretization (1D only). Classic numerical fluxes: Euler, Lax-Friedrich, Upwind, Lax-Wendroff. Absolute stability and CFL condition. Von-Neumann stability analysis.

7)      First order sytems of hyperbolic equations. Solution as superposition of waves. Numerical treatment of boundary conditions.

Course organization.

The course is organized in taught lectures complemented by exercise sessions where, with the help of computer codes, the students may experiment themselves the proposed numerical schemes and apply them to practical problems. The student assessment consists of a written examination, with theoretical questions and a practical exercise, possibly followed by an oral examination.

 


Note Sulla Modalità di valutazione

Course organization.

The course is organized in taught lectures complemented by exercise sessions where, with the help of computer codes, the students may experiment themselves the proposed numerical schemes and apply them to practical problems. The student assessment consists of a written examination, with theoretical questions and a practical exercise, possibly followed by an oral examination.

 


Bibliografia

Software utilizzato
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Aule
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FreeFem++ Vedi sito produttore SI SI
MATHWORKS Matlab SI SI

Mix Forme Didattiche
Tipo Forma Didattica Ore didattiche
lezione
45.0
esercitazione
0.0
laboratorio informatico
24.0
laboratorio sperimentale
0.0
progetto
0.0
laboratorio di progetto
0.0

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
schedaincarico v. 1.6.9 / 1.6.9
Area Servizi ICT
05/12/2021