37261 - Numerical Analysis

Learning outcomes

The course covers the basic techniques for iterative methods for solving linear system of equations, the solution of problems modeled by Ordinary Differential Equations (ODE) e BVP (Boundary Value problems). Next, the course presents numerical methods for the solution of problems modeled by Partial Differential Equations (PDE).

Course contents

Prerequisites:

A prior knowledge and understanding of Mathematical Analysis, Geometry and Matlab programming are required.

Program:

1. Iterative Methods for the numerical solution of linear and non-linear systems.

2. Numerical Integration: Gaussian quadrature.

3. Numerical Differentiation: derivative approximation by Finite Differences.

4. Ordinary Differential Equations: Initial Value Problems. One-step Methods (Runge-Kutta) and Adams Multistep Methods. Stiff problems. BDF Methods.

5. Boundary-Value Problems for Ordinary Differential Equations: Shooting Methods; Finite Difference Methods; Collocation Methods.

6. Partial Differential Equations. Some classical examples of problems described by partial differential equations. Classification. Strong and weak formulations. Hyperbolic, parabolic and elliptic linear problems. Numerical methods for their resolution: Finite Difference methods, Galerkin methods. Finite Element methods, applied to both ordinary and partial differential linear equations.

Readings/Bibliography

It will be fundamental to use the notes taken during the lectures and the computer material made available on the web. For further study we recommend:

[1] C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995.

[2] R.J. LeVeque, Finite Difference Methods for ODEs and PDEs, Steady State and Time Dependent Problems. SIAM, Philadelphia, 2007.

[3] U.M. Ascher, L.P.Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Siam 1998

[4] M.S. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM 2006.[5] A. Quarteroni, Numerical Models for Differential Problems, 2nd ed., Springer 2014.
[6] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer, 2000.

Teaching methods

The course is structured in lectures and exercises in the computer laboratory. More precisely, the lectures on the numerical methods for differential problems described by ordinary or partial differential equations, are followed by laboratory exercises aimed at implementing these methods in MATLAB and developing an adequate sensitivity and awareness of their use.

Assessment methods

The exam aims to verify the achievement of the following educational objectives:

- knowledge of the numerical-mathematical aspects and of the main algorithmic methodologies that deal with the numerical solution of differential problems described by ordinary or partial differential equations;

- ability to solve real problems of interest in Engineering by using or developing numerical methods and writing the corresponding algorithms in MATLAB.

The end-of-course exam (the evaluation of which is in thirtieths) will take place in a single test which includes both the development of MATLAB codes for the numerical solution of differential problems, and the written answer to theoretical questions on the topics covered in the lessons.

During the test, the use of support material such as textbooks, notes, computer supports is not allowed.

Teaching tools

The course includes a laboratory activity in which the MATLAB software will be used. The corresponding teaching material will be made available to the student in electronic format and will be available on the lecturer's website.

Office hours

See the website of Lucia Romani