B012965 - NUMERICAL MODELS FOR SIMULATION

Italian or English (depending whether there are students from abroad)

Difference equations, stability of solutions, linear multistep methods for ordinary differential equations. Functions of matrices, sequences of functions of matrices, positive matrices. Linear systems. Nonlinear systems, linearization, Liapunov's functions. Runge-Kutta methods. Conservative problems. Applications.

- L. Brugnano. Modelli Numerici per la Simulazione. MasterBooks, Firenze, 2018.

Knowledge acquired:

Knowledge about both continuous and discrete dynamical systems, and basic numerical methods for differential equations.

Skills acquired (at the end of the course):

Ability to analyze and simulate dynamical systems, both continuous and discrete.

Functions of several variables; linear algebra; Matlab programming.

Lectures

Teaching tools:

http://web.math.unifi.it/users/brugnano/Corsi/index.htm

By selecting the academic year, it is possible to download the lecture notes.

For reception see the web page:

http://web.math.unifi.it/users/brugnano/Corsi/ORARIO%20RICEVIMENTO.htm

Address:

Viale Morgagni, 67/a - 50134 Firenze
Tel: 055 2751-421

E-mail: luigi.brugnano@unifi.it
Web: http://web.math.unifi.it/users/brugnano/

The exam consists in an oral dissertation, concerning the methodological aspects of the course, plus a a written report concerning the efficient Matlab implementation of the methods and models studied. The written report is done by the student before the exam, alone or in a group of 2 people. The final score is obtained as the combination of the oral dissertation (with weight 2/3) and of the written report (with weight 1/3) scores.

• Difference equations: preliminary notions, the difference and shift operators, factorial powers, particular cases, comparison principle.
• Linear difference equations: general solution, the constant coefficients case, stability of solutions, cobweb model in economy amd model of economy of a nation, linear multistep methods, consistency, zero-stability, and convergence, absolute stability, Dahlquist's barriers.
• Functions of matrices: minimal polynomial, functions of matrices, component matrices, sequences of functions of matrices, analysis through the Jordan canonical form, positive matrices, theorem of Perron-Frobenius.
• Linear systems: linear systems of ordinary differential equations and linear systems of difference equations, model of arms race, stiffness of a linear problem and role of A-stable methods.
• Nonlinear systems: nonlinear systems of difference equations and nonlinear systems of ordinary differential equations, linearization process, Liapunov functions, applications. Generalization of the concept of stiffness for nonlinear problems.

(End of program for the course of Approximation Methods)

• Examples of nonlinear models: the predator-prey model, the logistic equation, mention about chaotic dynamics.

• Runge-Kutta methods: order and linear stability analysis. Hamiltonian problems and HBVM methods.