Il corso è tenuto in inglese. Il corso introduce i metodi numerici per la risoluzione di equazioni differenziali ordinarie e alle derivate parziali, affrontando sia gli aspetti teorici sia quelli applicativi. L'obiettivo principale è fornire agli studenti le conoscenze e le competenze pratiche necessarie per eseguire simulazioni numeriche efficienti di problemi classici dell’ingegneria, una competenza essenziale per corsi avanzati e per progetti di laurea magistrale. Un aspetto centrale del corso è quello di fornire agli studenti una solida conoscenza delle metodologie nel calcolo numerico e addestrarli all’uso del software Matlab, affinché siano in grado di risolvere numericamente, con l'ausilio del calcolatore, alcuni problemi che comunemente si presentano nelle applicazioni dell’ingegneria. In particolare, tali conoscenze consentono agli studenti di individuare tra i metodi a loro disposizione quello più efficiente, sia in termini di complessità computazionale che di occupazione di memoria, per la risoluzione di un determinato problema, e di fare un’analisi critica della soluzione ottenuta. Questa capacità è particolarmente importante per gli ingegneri chimici che si troveranno a utilizzare codici commerciali per risolvere problemi complessi, come ad esempio quelli connessi con i processi di trasformazione dei materiali. Infatti, grazie a questo insegnamento, gli studenti acquisiscono le competenze matematiche necessarie per evitare risultati non attendibili o comunque poco accurati, che potrebbero derivare da scelte non adeguate degli algoritmi.

By the end of the course, students will be able to identify and classify Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) underlying mathematical models of engineering problems, and they will acquire foundational knowledge of finite difference and finite element methods for solving such equations. In addition, students will develop the skills to design and implement efficient numerical methods for ODE/PDE-based engineering problems using Matlab.

Basic notions of calculus and numerical linear algebra are required, along with familiarity with basic programming constructs and the Matlab software.

INTRODUCTION (4 hours) Lectures: Generalities on ordinary and partial differential equations. Examples of classical differential problems, in particular with reference to the chemical engineering ones. Stationary problems, eigenvalue problems and evolution problems. Classification of second order equations. Initial-value and boundary-value problems. Strong and weak solutions. FEM FOR LINEAR STATIONARY PROBLEMS (40 hours) Lectures and problem sessions (28 hours): • The Finite Element Method (FEM) for second order linear elliptic problems defined on bounded intervals or on bounded plane domains. • Sobolev spaces and weak formulations of differential problems. Integration by parts and Green’s formulas. • Problem space domain discretization. Finite elements and finite element spaces. Definitions and examples. Some main results on approximation theory. • Application of the FEM to linear elliptic problems, first defined on bounded intervals, and then on bounded plane domains, with different types of boundary conditions (Dirichlet, Neumann, Robin, mixed-type, of periodicity). • Some basic convergence results. • Eigenvalue problems. • Construction of the FEM stiffness matrix and solution of the associated linear system. • Numerical integration formulas. Conditioning of a linear system. Basic methods for the numerical solution of linear systems. Computer Lab sessions (12 hours) FEM FOR LINEAR TIME-DEPENDENT PROBLEMS (16 hours) Lectures and problem sessions (10 hours): • Weak formulations in the space domain of parabolic and hyperbolic problems, in particular of transport and wave propagation problems. Application of the FEM. • Basic on initial value ODE problems. • Finite difference schemes for the time discretization of ODE systems generated after performing the (space) FEM discretization. Computer Lab sessions (6 hours) NUMERICAL METHODS FOR ODE INITIAL-VALUE PROBLEMS (IVP) (10 hours) Lectures and problem sessions (7 hours): • One-step methods and convergence results for the solution of IVP-ODEs. • Stiff problems and numerical methods for their resolution. Computer Lab sessions (3 hours) FINITE DIFFERENCE METHODS FOR PDE BOUNDARY-VALUE PROBLEMS (BVP) (10 hours) Lectures and problem sessions (7 hours): • Finite difference methods for elliptic, parabolic and hyperbolic BVP-PDEs. • Stability and convergence results. • Reaction-diffusion-transport problems and numerical stabilization for transport dominating ones. Computer Lab sessions (3 hours)

The above lecture contents include also corresponding exercise sessions, for a total of 21 hours. Important aspects of the lecture topics as well as the solution of some problems will be discussed, to help the students to better understand the lectures. An additional computer room activity (24 hours) is also scheduled. This activity focuses on solving engineering problems using the MATLAB programming language, both by implementing algorithms introduced during the lectures and by using Matlab toolboxes. Students will explore the numerical properties of the methods discussed and will be guided through a critical analysis of the computed results

Lecture notes, as well as notes on the computer lab sessions, can be downloaded by the students from the Portale della Didattica. Further reading: A. Quarteroni, Numerical Models for Differential Problems, Springer Verlag, 2009. A. Quarteroni, F. Saleri, P. Gervasio, Scientific Computing with MATLAB and OCTAVE, Springer, 2014.

Slides; Dispense; Esercizi; Esercizi risolti; Esercitazioni di laboratorio; Esercitazioni di laboratorio risolte; Strumenti di simulazione; Strumenti di auto-valutazione;

Modalità di esame: Test informatizzato in laboratorio; Prova scritta (in aula);

Exam: Computer lab-based test; Written test;

... The exam consists of two tests: the first is an computer test, the second is a written test. The first test, to be held in the computer Lab, consists of 4 multiple-choice questions concerning the course topics. In this test students must show to be able to solve, by applying the numerical methods described in the lectures and by preparing short Matlab scripts, problems similar to those that have been solved during the Lab course sections. This to verify the students’ knowledge of the algorithms presented during the course and their programming and computing skills. Students must use the Matlab software installed in the Lab PC, and they are not allowed to use books, notes, personal PC and any electronic device. The maximum grade is 16 points: each correct answer is worth 4 points, an unanswered question is worth 0 points, and an incorrect answer results in a 20% penalty. The Lab test is scheduled for about an hour and a half. The second written test, to be held in a lecture room the same day of the Matlab test, consists of two/three exercises that include both practical and theoretical questions, covering the entire course content. The written test is scheduled for about an hour and a half and is worth up to 16 points. During the exam, it is forbidden to use books, personal material such as notes and handouts and any electronic device. The final grade is the total of the two test scores. A minimum score of 18 is required to pass, while full marks with all correct answers are awarded with honors (cum laude).

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.