Numerical Modelling of Differential Problems
The course aims at providing the fundamental tools for the numerical simulation of problems governed by partial differential equations. In particular the course addresses elliptic, parabolic and hyperbolic equations and methods for their numerical solution by finite differences, finite volume and finite elements. The issues of consistency, stability and convergence of the methods as well as their practical implementation are analysed in detail.
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.
The course is organized in taught lectures complemented by exercise sessions where, with the help of computer codes, the students may experiment themselves the proposed numerical schemes and apply them to practical problems. The student assessment consists of a written examination, with theoretical questions and a practical exercise, possibly followed by an oral examination.