2018/2019

2020/2021

DM270

MAT/08 (NUMERICAL ANALYSIS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

3°

1st semester (01/10/2020 - 20/01/2021)

6

56 lesson hours

Italian

WRITTEN AND ORAL TEST

MOIOLA ANDREA (titolare) - 6 ECTS

The skills aquired in the Numerical Analysis class and basic knowledge of the MATLAB language.

The course aims to integrate and extend the knowledge gained in the previous numerical analysis courses.
The fundamental objective is to present several techniques of numerical modelling for the solution of boundary value problems, mostly of diffusion-advection-reaction type.
We will restrict to the one-dimensional case, but the ideas and the techniques learned may be applied to much more general situations.

We introduce numerical algorithms for the solution of differential boundary values problems.
Elements of the MATLAB language are part of the course content.

- Shooting method for linear and non-linear boundary value problems.
- Diffusion-transport-reaction models.
- Well-posedness of the Dirichlet problem in one dimension, maximum principle, Green function, other boundary conditions.
- Numerical differentiation: finite differences; truncation error and round-off error.
- Finite difference method.
Existence, uniqueness and accuracy of the discrete solution.
The Neumann problem.
Efficient implementation.
Diffusion-transport problem, upwind method.
Eigenvalue problems.
Non-linear problems.
Uncertainty quantification.
- Polynomial and trigonometric collocation method; the discrete Fourier transform and the FFT.
- The weak formulation of a boundary value problem; abstract variational problems.
- The Galerkin method.
- The finite element method; linear and quadratic finite elements; error analysis.
- Evolution problems: the heat equation, the Fourier method, the theta-method.

Lectures and tutorials in the computer lab.

Lecture notes prepared by the lecturer.

Further references:
V. Comincioli, Analisi Numerica. Metodi, Modelli, Applicazioni, McGraw-Hill, 1995.
A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2009.
R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-state and Time-dependent Problems, SIAM 2007.
A. Quarteroni, Modellistica Numerica per Problemi Differenziali, Springer, 2016.
A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, Matematica Numerica, Springer, 2014.
G. Strang, G. Fix, An Analysis of the Finite Element Method, Wellesey–Cambridge press, 2008 (1st ed. 1973).
E. Suli, D. Mayers, An introduction to Numerical Analysis, Cambridge University Press, 2003.
A. Tveito, R. Winther, Introduction to Partial Differential Equations. A Computational Approach, Springer 2005.

Written and oral exam with discussion of MATLAB reports.
The students will have to show that they know the theoretical concepts, are able to apply them to concrete problems, can compare different strategies, can describe, implement and analyse the numerical schemes that constitute the course programme.

If for health reasons it will be impossible to take the exam on site, the remote exam might be done in oral form.