The course aims to integrate and extend the knowledge gained in the previous numerical analysis courses, with particular attention to the solution of initial and boundary value problems. The fundamental objective is to present several techniques of numerical modelling, both revisiting the classic algorithms of numerical analysis and introducing new methods of approximation.

Lectures and tutorials in the computer lab

Written and oral exam with discussion of MATLAB reports

Numerical analysis 1 and 2, and basic knowledge of the MATLAB language.

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

Initial value problems:
- one-step numerical algorithms for the solution of the Cauchy problem;
- multistep methods;
- high-order one-step methods;
- convergence of one-step methods;
- 0-stability, consistency and convergence of multisteps methods;
- absolute stability, stiff problems.

Boundary value problems:
- diffusion models, existence and uniqueness of the solution of the problem of diffusion-reaction with boundary conditions of Dirichlet and Neumann type;
- numerical methods for solving boundary value problems, shooting method;
- finite difference method, existence and uniqueness of the solution of the discrete problem of diffusion-reaction;
- advection-diffusion problem, upwind scheme;
- evolutionary problems, heat equation, theta-method;
- variational methods, weak formulation of the problem of diffusion-reaction;
- bilinear forms and Galerkin approximation method;
- consistency and convergence of the Galerkin method.

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio: "Matematica Numerica", Springer, 2014.
V. Comincioli: "Analisi Numerica, metodi, modelli, applicazioni", McGraw-Hill, 1995.
A. Quarteroni: "Modellistica Numerica per problemi differenziali", Springer, 2016.