NUMERICAL METHODS FOR DIFFERENTIAL AND INTEGRAL EQUATIONES
Learning outcomes of the course unit
Knowledge: basic concepts for the numerical modeling of partial differential equations, fundamental elements of the finite differences techniques, finite elements techniques, spectral methods and the BEMethod; elementary concepts of mathematical optimization.
Skills: application of knowledge to the classical linear elliptic and parabolic equations with the acquisition of autonomy in the evaluation of algorithmic aspects of implementation related to stability and efficiency.
Knowledge of a programming language (Matlab, Fortran).
Course contents summary
Problems for equations of elliptic type: variational formulation of elliptic boundary value problems; collocation method, 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.
• “Problemi e Modelli Matematici nelle Scienze Applicate”, V. Comincioli, ed. CEA, 1993.
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 supervised autonomous re-elaboration 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.