Ing Ind - Inf (Mag.)(ord. 270) - BV (469) AERONAUTICAL ENGINEERING - INGEGNERIA AERONAUTICA

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096010 - NUMERICAL MODELING OF DIFFERENTIAL PROBLEMS

Obiettivi dell'insegnamento

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.

Risultati di apprendimento attesi

The students are expected to:

- be able to apply and implement classical numerical methods based on finite element for the solution of standard elliptic, parabolic and Navier-Stokes equations and finite volume methods for the solution of linear hyperbolic equations;

- be able to characterize the properties of numerical methods in terms of consistency, convergence and stability;

Argomenti trattati

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.

Prerequisiti

The students are assumed to be familiar with:

- linear algebra and basic calculus;

- numerical methods for solving linear systems;

- numerical methods for solving nonlinear equations;

- numerical methods for solving Ordinary Differential Equations.

Modalità di valutazione

The student assessment consists of a written examination, with theoretical questions and a practical exercise, possibly followed by an oral examination.

The students are expected:

A) Knowledge and understanding: analyze and characterize the properties of numerical methods for the solution of PDEs.

B) Applying knowledge and understanding: choose the proper numerical methods for the solution of different PDEs and implement the method in a computer code.

The students are not expected to be able to solve only standard exercises but to be able to apply methods using a critical thinking approach. They are also expected to be able to expose in a clear way the main most important theoretical results.

Bibliografia

Alfio Quarteroni, Numerical Models for Differential Problems, Editore: Springer, Anno edizione: 2017, ISBN: 978-3-319-49315-2

Forme didattiche

Tipo Forma Didattica

Ore di attività svolte in aula

(hh:mm)

Ore di studio autonome

(hh:mm)

Lezione

39:00

58:30

Esercitazione

21:00

31:30

Laboratorio Informatico

0:00

0:00

Laboratorio Sperimentale

0:00

0:00

Laboratorio Di Progetto

0:00

0:00

Totale

60:00

90:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione

Insegnamento erogato in lingua
Inglese

Disponibilità di libri di testo/bibliografia in lingua inglese

Possibilità di sostenere l'esame in lingua inglese