# NUMERICAL MODELS AND METHODS

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Versione italiana
2016/2017
Teacher
LORENZO PARESCHI
Credits
6
Didactic period
Secondo Semestre
SSD
MAT/08

#### Training objectives

The goal of the course is to present the basis of mathematical modelling with differential equations and of numerical methods for such models. The course includes a part of hours dedicated to the laboratory activities , in which it makes use of Matlab.

The main knowledge provided by the course will be :

- General introduction to partial differential equations
- Introduction to finite difference methods for partial differential equations
- Analysis of the stability and convergence of the main methods
- Extension of the results to multiple dimensions and boundary conditions

The main skills that students must acquire ( ie 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 ;
- Knowing how to write Matlab code that allows you to calculate the solution of a problem involving partial derivatives .

#### Prerequisites

To follow the course a full knowledge of the basics of numerical analysis is recommended. Knowledge of Matlab language is highly recommended and a great help.

#### Course programme

The course includes 48 hours (36 hours classroom on the theoretical aspects and a 12-hour lab sessions using Matlab).

1. Introduction to partial differential equations: definitions and first examples
2. Numerical methods: method of lines, Runge-Kutta methods, IMEX schemes
3. Diffusion equations: finite difference methods, finite elements methods, spectral methods, application to heat propagation
4. Transport equations: finite differences, intepolation methods, application to diffusion of pollutants in the air
5. Hyperbolic equations: finite difference methods, shock waves, high resolution methods, central schemes, application to traffic flow

#### Didactic methods

The course is based on theoretical lectures on all the topics of the course program (36 hours) and computer session to implement the various algorithms and methods with Matlab and to test them on simple problems (12 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

- Numerical Solution of Partial Differential Equations. An Introduction.
2nd Edition. K. W. Morton, D. F. Mayers, Cambridge University Press, 2005
- Matlab concetti e progetti, Lorenzo Pareschi, Giovanni Naldi, Apogeo, 2007
- Teacher notes