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.

TOPICS

First Part – Differential modeling and finite difference approximation

1. Review

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.

1. 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.

2. Diffusion

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.

3. 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.

4. 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.

5. Basics of Functional Analysis

Mathematical Methods: Lebesgue integral. Projection theorem and Riesz representation theorem. Schwartz distributions. Sobolev spaces.

6. 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.

7. 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.

There are five examination dates (two in January-February, two in June-July, one in September). The course consists in two moduli, one in Mathematical Methods (ref. G. Arioli and G. Grillo) and one in Numerical Methods (ref. C. Vergara and P. Zunino).
The exam in Mathematical Methods is written and consists in both questions on the theory and exercises. The exam in Numerical Methods consists in a written part and an optional oral part. The written part of both moduli takes place in the same day. Students can take the oral exam for the Numerical Methods modulus only when the corresponding written grade is at least 15 out of 30.

The final grade is the (rounded up) arithmetic mean of the grades obtained in the two moduli. To get the grade “30 cum laude” one should obtain such grade in both the subparts. If this is not the case, a single "30 cum laude" in one of the parts will be considered as 30 in performing the arithmetic mean.

It is possible to take the exam in one modulus (Mathematical or Numerical Methods) in one of the five examination dates and in the other modulus in another examination date, provided such dates are in the same academic year. It is mandatory to take the written and the oral part of the Numerical Methods modulus within the same examination date, i.e. it is not possible to give the written part in one examination date and the oral part in another one.

The participation to the written exam of one of the two moduli, automatically discards any previous grade obtained for that modulus, even if the student chooses to withdraw.

NOTES

1)  Mathematical Methods in Engineering and Numerical Methods in Engineering are also single courses which can be taken independently.

2)  Browsing texts, notes, and electronic devices are not allowed during the tests. It is mandatory to bring an ID (e.g. identity card, driver’s licence,…) in order to be identified.

3)  Registration to the exam is mandatory. Unregistered students will not be admitted.