Lecturer: Luisa Donatella Marini  

Course name: Numerical Methods In Engineering Sciences
Course code: 504710
Degree course: Ingegneria per l'Ambiente e il Territorio, Ingegneria Elettrica, Computer Engeneering
Disciplinary field of science: MAT/08
The course relates to:
University credits: ECTS 6
Course website: http://www.imati.cnr.it/marini

Specific course objectives

The aim of the course is to enable students to classify real-life problems and choose the best suited algorithms for dealing with them, in terms of costs/benefits and convergence properties. At the same time, the course is meant to make students well acquainted with the use of Matlab software and with the practical implementation of some algorithms.

Course programme

The course is divided in two parts, devoted essentially to the numerical approximation of boundary value problems for Partial Differential Equations (Pde's), and of initial value problems for Ordinary Differential Equations (Ode's). The basic common and necessary instruments to deal with both classes of problems are also developed.

Numerical solution of boundary value problems for Partial Differential Equations (Pde)

• Finite Difference method on a model problem in 1D.
• Consistency and Stability - Lax's Theorem for convergence of a numerical scheme.
• Finite Element method on a model problem in 1D: Variational formulation, continuous piecewise linear finite element approximation, stability and convergence; construction of the final system and comparison with finite differences.
• Finite Element method on a model problem in 2D: Functional spaces H1 and H10, energy norm; Variational Formulation, Lax-Milgram Lemma; Continuous piecewise linear finite element discretization on triangular meshes; Stability and convergence; Explicit computation of the elementary stiffness matrix and right-hand side; Assembling and solution of the final system.
• Various examples of boundary value problems in 2D.

Numerical solution of initial value problems for Ordinary Differential equations (Ode)

• One-step methods: Euler backward and forward, Crank-Nicolson, Heun; Stability and A-stability, consistency, convergence and order of convergence.
• Multistep Methods: general structure, consistency and stability conditions; Explicit and Implicit Adams methods.
• Runge-Kutta methods: consistency and stability conditions; example of construction of an explicit RK-method (Hints on predictor-corrector methods).
• Systems of Ordinary Differential Equations: stiff problems.

Common tools
The following list gives the necessary tools to tackle the two above classes of problems.

Reminders on Linear Algebra

• Nonlinear equations: bisection and Newton's methods. Convergence, order of convergence, stopping criteria.
• Nonlinear systems of equations: Newton's method and variants.

Approximation of functions and data

• Lagrange interpolation: interpolation error, piecewise Lagrange interpolation, order of approximation in various norms.
• Least squares method for data fitting: linear regression and various examples.

Numerical integration

• Interpolatory quadrature formulas in 1D: midpoint, trapezoidal, Simpson and error analysis. Gaussian formulae.
• Extension to dimension 2 on rectangular domains. Quadrature formulas on triangular domains: barycenter, vertex, and midpoint of the edges.

Course entry requirements

Differential and integral calculus for real functions; complex numbers; linear algebra; computer programming experience.

Course structure and teaching

Lectures (hours/year in lecture theatre): 45
Practical class (hours/year in lecture theatre): 0
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

A. Quarteroni, R. Sacco, F. Saleri . Numerical Mathematics-2nd edition. Springer Series: Texts in Applied Mathematics, Vol. 37 (2007)..

Testing and exams

The course includes two activities, strictly related to one another:
1) theoretical classes;
2) computer laboratory for the implementation of some of the numerical methods discussed in the classroom.
Before the end of the semester students must submit for evaluation a written dissertation discussing the results obtained during practical classes. Students are encouraged to work by groups of 3-4. The evaluation results will be part of the final grade.
Accordingly, the exam consists in a written part and an oral part.
Written part: a dissertation describing the results obtained in the practical classes.
Oral part (not compulsory for students who had a positive grade in the written part): a discussion on subjects developed during the theoretical classes. It is intended to verify the individual preparation.
Note: In absence of the written dissertation the oral exam is compulsory, and the maximum obtainable grade is 24/30.