NUMERICAL METHODS FOR DIFFERENTIAL AND INTEGRAL EQUATIONES
Learning outcomes of the course unit
- Knowledge and understanding of elementary concepts for the numerical modeling of partial differential differential problems and of the basis for the application of collocation method, finite differences, finite elements and spectral methods and of the boundary element method.
- Ability to apply knowledge and understanding, through mathematical programming in Matlab, to classical elliptic and parabolic linear equations with acquisition of autonomy in the evaluation of algorithmic implementation aspects regarding stability and efficiency.
- Autonomy of judgment in evaluating the approximation algorithms and the obtained results also through discussion with one's peers in possible teamworks.
- Ability to clearly communicate the acquired concepts and to discuss the obtained results.
- Ability to learn the drawbacks and the advantages of models and methods of resolution and to apply them in different working and scientific contexts.
Basic methods and algorithms of numerical analysis. Knowledge of a programming language.
Course contents summary
Problems for equations of elliptic type: collocation method; variational formulation; Galerkin method (finite elements, spectral elements) and hints to the Boundary Element Method (BEM); finite difference method and stabilization methods for advection-diffusion-reaction problems. Approximation of evolutionary problems for equations of parabolic type: semi-discretization in space and time, teta-method, finite difference method.
• “Modellistica Numerica per Problemi Differenziali”, A. Quarteroni, ed. Springer, 2000.
• “Numerical Approximation of Partial Differential Equations”, A. Quarteroni, A. Valli, ed. Springer, 1994.
During the lectures the contents of the course will be analyzed, highlighting the difficulties related to the introduced numerical techniques. Moreover, the course will consist of a part of autonomous re-elaboration, supervised by the professor, consisting in the application of the numerical techniques through laboratory programming. This activity will allow students to acquire the ability to deal with "numerical" difficulties and to evaluate the reliability and consistency of the obtained results.
Assessment methods and criteria
The exam includes:
• the assignment of a work for the application of numerical techniques introduced to solve a specific problem. The analysis of the results obtained by the student will allow to evaluate the acquisition of the above listed skills. In particular the threshold of sufficiency is fixed to the ability to achieve reliable numerical results.
• an assessment of the knowledge through a discussion of topics of the course. The threshold of sufficiency consists in the knowledge of the discriminating characteristics of the various methods presented in the course.