Numerical Modeling of Differential Problems

Synthetic program:

MODELLISTICA NUMERICA PER PROBLEMI DIFFERENZIALI: il corso si propone di fornire strumenti matematico numerico di utilizzo pratico per la risoluzione di problemi governati da equazioni differenziali alle derivate parziali. Il corso sarà coadiuvato sia da esercitazioni pratiche che da attività in laboratorio informatico dove lo studente avrà modo di sperimentare ed implementare direttamente le tecniche studiate. Lo studente apprenderà metodi per la modellistica numerica di equazioni alle derivate parziali quali elementi finiti, volumi finiti e differenze finite e verrà introdotto alla analisi di convergenza di tali schemi. Si affronteranno metodi numerici per problemi ellittici, parabolici, iperbolici lineari e di punto sella, con particolare riguardo ad applicazioni in campo aeronautico.

1) Short introduction to functional analysis: linear spaces, Hilbert spaces, Sobolev spaces. Concept of internal product and norms. Fundamental inequalities.
2) Elliptic problem: Laplace equation, convection-diffusion equations.Weak formulation, Galerkin and finite element discretization and the resulting algebraic system. General results of consistency, stability and convergence of the method. Stabilization techniques for convection dominated problems.
3) Parabolic equations. Weak formulation and finite element discretization. Integration in time. Main convergence results.
4) Stokes problem. Compatible boundary conditions. Weak formulation. The pressure-velocity coupling: stability condition. Discretization by finite elements. Compatible ad incompatible finite element spaces. Stabilization techniques. The saddle point algebraic problem.
5) Incompressible Navier-Stokes. The different treatment of the convective term: implicit, semi-explicit, fully explicit. Finite element discretization. Fixed point techniques for the nonlinear term. Semi-Lagrangian schemes. Fractional step methods: Chorin-Temam scheme in its basic and incremental form. Stabilization techniques.
6) First order linear hyperbolic equations. Boundary conditions. The method of characteristics. Finite volume discretization (1D only). Classic numerical fluxes: Euler, Lax-Friedrich, Upwind, Lax-Wendroff. Absolute stability and CFL condition. Von-Neumann stability analysis.
7) First order sytems of hyperbolic equations. Solution as superposition of waves. Numerical treatment of boundary conditions.