Lecturer: Elena Bonetti   Ugo Pietro Gianazza  

Course name: Modelli matematici e calcolo numerico
Course code: 502558
Degree course: Ingegneria per l'Ambiente e il Territorio
Disciplinary field of science: MAT/08 MAT/05
L'insegnamento costituisce attività di base per: Ingegneria per l'Ambiente e il Territorio
University credits: CFU 9
Course website: http://www.imati.cnr.it/~gianazza/mod_calc.html

Specific course objectives

The aim of the course is twofold. On one hand at the end of the course the student should be able to write simple mathematical models of some physical phenomena, which are particularly interesting under the point of view of engineering applications; on the other hand the student should be able to apply numerical algorithms, in order to solve these same models.

Course programme

The first five topics concern the Numerical Analysis part, whereas the remaining arguments are more related to modelling and Mathematical Analysis.

Nonlinear equations/systems

• Bisection and Newton methods, convergence and order of convegence
• Stopping criteria

Approximation of functions and data

• Lagrange interpolation, global and piecewise
• Convergence analysis
• Least square approach for the data-fitting

Numerical Derivation and Integration

• Approximation of derivatives
• Basic integration formulae
• Error analysis and practical use

Solution of linear systems of equations

• Direct methods (Gaussian elimination and LU factorization, Choleski factorization), implementational aspects and costs
• Iterative methods (Jacobi and Gauss-Seidel), convergence analysis, implementational aspects, stopping criteria

Ordinary differential equations/systems

• One-step methods (Euler backward and forward, Crank-Nicolson, Heun)
• Consistence, 0-stability and A-stability, convergence and orders of convergence.
• Computational aspects and application to systems

Differential models

• Introduction.
• Well-posed problems.
• Integration by parts formulae.

Ordinary differential equations

• Cauchy Problems.
• Differential linear and nonlinear systems.
• Autonomous systems.
• Asymptotic behavior, equilibrium points.
• Stability in the sense of Liaponouv.

Partial differential equations (of second order)

• Diffusion phenomena: the heat equation.
• The Laplace equation.
• Waves and vibrations: the wave equation.
• The Navier-Stokes equation.

Functional Analysis methods for differential equations

• Hilbert spaces.
• Linear operators and linear functionals.
• Weak formulation of steady state and evolution problems.

Course entry requirements

Differential and integral calculus for real functions, complex numbers, vector and matrix calculus.

Course structure and teaching

Lectures (hours/year in lecture theatre): 52
Practical class (hours/year in lecture theatre): 31
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

Lecture notes properly prepared.

A. Quarteroni, F. Saleri. Calcolo Scientifico - IV Edizione. Springer-Verlag Italia, Milano 2008.

S. Salsa, F.M.G. Vegni, A. Zaretti, P. Zunino. Invito alle equazioni alle derivate parziali – Metodi, modelli e simulazioni. Springer-Verlag, Milano, 2009.

Testing and exams

The exam consists in a written test and an oral examination, the latter one conditioned by the outcome of the former one.