GOALS AND CONTENTS OF THE COURSE
The goal of the course is twofold. First, we present some classical differential models of Mathematical Physics, developing and analyzing analytical methods for the computation of the solutions and some finite difference schemes for their approximation. Second we introduce the variational formulation of some boundary value problems, together with the finite element method for their numerical approximation. The course is characterized by a constant synergy between modeling, theoretical aspects and numerical simulation.
First Part – Differential modeling and finite difference approximation
Mathematical Methods: Differential calculus for functions of several real variables. Series of functions.
Numerical Methods: Finite difference formulae to approximate derivatives. Numerical approximation of ordinary differential equations, convergence, absolute stability.
- First-order conservation laws
Mathematical Methods: Transport equation. Traffic flow models. Method of characteristics. Rankine-Hugoniot relation. Shock and rarefaction waves. Entropy condition.
Numerical Methods: Approximation with finite differences. Convergence, consistency, zero-stability and absolute stability. Forward Euler-centered scheme. Upwind,Lax-Friedrichs and Lax-Wendroff schemes. Analysis of the schemes, CFL condition and its meaning. Backward Euler-centered scheme. A quick description of systems and of non-linear problems.
Mathematical Methods: Heat equation. Well-posed problems. Separation of variables. Maximum principles. Fundamental solution. Cauchy problem in the half-space. Duhamel principle.
Numerical Methods: Discretization of the heat equation with finite differences. Implicit and explicit time marching schemes, the theta-method, stability analysis.
- Laplace-Poisson equation
Mathematical Methods: Harmonic functions. Mean value properties. Maximum principles. Well-posed problems. Poisson’s formula for the disk. Newtonian potentials.
Numerical Methods: Discretization with finite differences of a one-dimensional elliptic problem. Imposition of the Dirichlet and Neumann boundary conditions. Algebraic formulation and matrix properties. Diffusion-convection and diffusion-reaction problems.
- Wave equation
Mathematical Methods: String equations. Well-posed problems and separation of variables. D’Alembert formula.
Numerical Methods: Discretization of the wave equation with finite difference explicit and implicit schemes. Leapfrog and Newmark schemes. Stability properties.
Second Part – Functional Analysis, variational formulations and discretizations via finite element method.
- Basics of Functional Analysis
Mathematical Methods: Lebesgue integral. Projection theorem and Riesz representation theorem. Schwartz distributions. Sobolev spaces.
- Weak formulation and Finite Elements approximation of stationary problems
Numerical Methods: Bilinear form, abstract variational problems and Lax-Milgram lemma. Variational formulation of elliptic problems and applications to transport-reaction-diffusion equations. Introduction to the Galerkin method for a one-dimensional elliptic problem. Consistency, stability and convergence. Cea' Lemma. The finite elements method. Linear and quadratic finite elements. Definition of Lagrangian basis functions, of composite interpolation and error estimates. Extension to the 2D case. Approximation of the diffusion-convection-reaction problem: comparison with the finite difference case and stability analysis. Stabilization with the upwind strategy and the mass lumping technique.
- Evolution problems
Numerical Methods: Approximation with the Galerkin method, the semi-discrete problem. Explicit and implicit time marching schemes, the theta-method. Stability properties. A quick description of finite elements for hyperbolic problems.