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Academic year
Didactic period
Secondo Semestre

Training objectives

The goal of the course is to present the basic concepts of the mathematical modeling through the use of partial differential equations together with an introduction to the numerical methods used for solving such models. The course includes a part dedicated to the laboratory activities, in which the students will make use of Matlab

The main knowledge provided by the course will be :

- General introduction to partial differential equations: elliptic equations, parabolic equations, hyperbolic equations, transport-diffusion equations, conservation laws and balance laws.
- Introduction to finite difference, finite volume and finite elements methods for partial differential equations. Convergence analysis and stability of the most important methods. Algorithmic and implementations aspects in multidimensions.

The main skills that students must acquire ( i.e. the ability to apply knowledge ) will be :
- Identify the different types of partial differential equations;
- Be able to assess which approach is more efficient for a given problem ;
- Be able to solve simple problems with partial differential equations using different methods ;
- Be able to write Matlab code that allows to compute the solution of a physical problem involving partial differential equations .


To follow the course a good knowledge of calculus and mathematical analysis is recommended. Knowledge of Matlab language is highly recommended.

Course programme

The course includes 63 hours (48 hours classroom on the theoretical aspects and a 15 hours lab sessions using Matlab).

1. Introduction to partial differential equations: definitions, classifications and first examples.

2. Diffusion equations: finite difference methods, spectral methods. Applications to heat transfer problem in one and two spatial dimensions.

3. Elliptic equations: finite difference methods, finite elements methods, spectral methods. Applications to structural mechanics in one and two spatial dimensions.

4. Linear hyperbolic equations: finite volume methods. Applications to diffusion of pollutants in the air in one and two spatial dimensions.

5. Non linear hyperbolic equations: shock waves, high resolution methods. Applications to traffic flow.

6. System of conservation laws and balance laws: finite volume methods. Applications to compressible Euler equations in one spatial dimension and to the shallow water equations.

Didactic methods

The course is based on theoretical lectures on all the topics of the course program (48 hours) and computer session to implement the various algorithms and methods with Matlab and to test them on simple problems (15 hours).

Learning assessment procedures

The exam is composed of two parts, a practical one and a theoretical one.

The practical part consists in an applied project chosen by the student between a list proposed by the teacher. The project should be realized using Matlab and the student should produce a short report on this. 

The evaluation of this project will be then supplemented by an oral test .
This oral test will cover all the topics seen in class during the course.

Reference texts

1) Modellistica Numerica per Problemi Differenziali. A. Quarteroni. Springer, 2008.
2) Finite Volume Methods for Hyperbolic Problems. R. J. LeVeque. Cambridge University Press 2002.
3) Numerical Solution of Partial Differential Equations. An Introduction.
2nd Edition. K. W. Morton, D. F. Mayers, Cambridge University Press, 2005
4) E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, 2009.
5) E. Godlewski, P.A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996.