34794 - Numerical Analysis (2nd cycle)

Learning outcomes

The student learns the numerical methods for differential problems. At the end of the course the student knows the numerical-mathematical aspects and the main algorithmic methodologies that deal with the numerical solution of differential problems of interest in Engineering.

Course contents

Prerequisites:

A prior knowledge and understanding of Geometry and Algebra, Mathematical Analysis and Matlab programming is required. Moreover, a prior knowledge of the basic topics of Numerical Analysis is also needed.

Program:

1. A review of root-finding methods for one equation in one variable: the bisection method, Newton's method, the secant method. Iterative methods for the numerical solution of systems of non-linear equations: Newton's method and Quasi-Newton methods.

2. A short review of polynomial interpolation in Lagrange form and of interpolatory quadrature formulas (especially Newton-Cotes quadrature formulas). Open and closed Gaussian quadrature formulas. Numerical algorithms for computing nodes and weights of Gauss-Legendre, Gauss-Lobatto and Gauss-Radau formulas.

3. Approximation of the derivatives of a function by finite differences. The method of undetermined coefficients. How to determine the step size that minimizes the total error associated with finite-difference derivative approximations. Increasing the accuracy of finite-difference derivative approximations by Richardson's extrapolation.

4. Ordinary Differential Equations (ODEs): Initial Value Problems. One-step methods (Runge-Kutta) and Adams multistep methods. Predictor-Corrector methods. BDF methods. Stiff problems.

5. Two-point Boundary Value Problems for a second order linear/non-linear ODE: shooting methods and finite difference methods.

6. A brief overview of iterative methods for the numerical solution of linear systems that arise from the finite difference method.

Readings/Bibliography

It will be fundamental to use the notes taken during the lectures and the computer material made available at https://virtuale.unibo.it/. For further study we recommend:

[1] J. Stoer, R. Bulirsch: Introduction to Numerical Analysis (3rd ed.), Springer, 2002.

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

[3] A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, 2nd ed., Springer, 2007.

[4] R.J. LeVeque: Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, 2007.

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

[6] K. Atkinson, W. Han, D. Stewart: Numerical Solution of Ordinary Differential Equations, John Wiley and Sons, 2009.

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 differential equations, are followed by laboratory exercises aimed at implementing these methods in MATLAB and developing an adequate sensitivity and awareness of their use.

In consideration of the type of activity and the teaching methods adopted, the attendance of this training activity requires the prior participation of all students in Modules 1 and 2 of Health and Safety training courses, [https://elearning-sicurezza.unibo.it/?lang=en] in e-learning mode.

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 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 downloadable at https://virtuale.unibo.it/.

Office hours

See the website of Lucia Romani