# Matematica | SCIENTIFIC COMPUTING

## Matematica SCIENTIFIC COMPUTING

 0522200045 DIPARTIMENTO DI MATEMATICA EQF7 MATHEMATICS 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/08 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 BEATRICE PATERNOSTER T DAJANA CONTE
ExamDate
CALCOLO SCIENTIFICO12/06/2023 - 09:00
CALCOLO SCIENTIFICO03/07/2023 - 09:00
Objectives
KNOWLEDGE AND UNDERSTANDING:
THE AIM OF THE COURSE IS THE THEORETICAL KNOWLEDGE AND CRITICAL ANALYSIS OF THE MAIN NUMERICAL METHODS FOR THE SOLUTION OF PROBLEMS MODELED BY ORDINARY DIFFERENTIAL EQUATIONS, TOGETHER WITH THE DEVELOPMENT OF THE CORRESPONDING MATHEMATICAL SOFTWARE.

THE AIM OF THE COURSE IS TO MAKE THE STUDENT CAPABLE TO
•SOLVE SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS (ALSO LARGE SYSTEMS) BY USING MATHEMATICAL SOFTWARE
•THEORETICALLY AND EXPERIMENTALLY ANALYZE THE PROPERTIES OF NUMERICAL METHODS FOR ODES: CONVERGENCE AND STABILITY
•CHOOSE THE MORE APPROPRIATE NUMERICAL METHOD TO SOLVE THE PROBLEM UNDER EXAMINATION
Prerequisites
THEORY OF ORDINARY DIFFERENTIAL EQUATIONS.
BASICS ON PROGRAMMING LANGUAGE MATLAB
Contents
NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS: LINEAR MULTISTEP METHODS, PREDICTOR CORRECTOR METHODS, RUNGE-KUTTA METHODS. ORDER, ERROR ESTIMATIONS, CONSISTENCY, CONVERGENCE, ZERO-STABILITY. THEORY OF WEAK STABILITY. STIFF SYSTEMS. STRUCTURE OF A VARIABLE STEPSIZE ALGORITHM. STARTING PROCEDURES. LOCAL TRUNCATION ERROR ESTIMATION. STRATEGIES FOR STEPSIZE CHANGING. MATHEMATICAL SOFTWARE EVALUATION. NUMERICAL APPROXIMATIONS THROUGH NON POLYNOMIAL BASIS. INTRODUCTION TO NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS. WEAK AND STRONG CONVERGENCE.EULERO-MARUYAMA METHOD.
Teaching Methods
THE TEACHING IS COMPOSED OF FRONTAL LESSONS AND EXERCISES.

THE FRONTAL LESSONS WILL PRESENT THE METHODOLOGIES AND THE ALGORITHMS THAT THEN, DURING THE EXERCISES, WILL BE CODED IN SCIENTIFIC CALCULATION ENVIRONMENTS AND TESTED ON TEST PROBLEMS OF INTEREST.

PART OF THE EXERCISES WILL BE DEDICATED TO PROJECT ACTIVITIES IN SMALL GROUPS, FOR THE PURPOSE OF DEVELOPING MATHEMATICAL SOFTWARE AND TESTING IT ON TEST PROBLEMS PROVIDED BY THE TEACHER, VERIFYING THE PROPERTIES OF ACCURACY, STABILITY AND EFFICIENCY OF THE METHODS. WORKING IN SMALL GROUPS ALSO AIMS TO GET STUDENTS USED TO GROUP WORK.

Verification of learning
THE FINAL EXAM CONSISTS IN THE DISCUSSION OF A PRACTICAL PART IN THE LABORATORY AND AN ORAL PART ON THE CONTENTS OF THE COURSE. THE PRACTICAL PARTY REGARDS THE USE OF THE MATHEMATICAL SOFTWARE DEVELOPED DURING THE TEACHING, TO BE APPLIED TO CERTAIN TEST PROBLEMS BASED ON ORDINARY DIFFERENTIAL EQUATIONS, TO CHECK THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE. THE ORAL PART REGARDS THE THEORETICAL CONTENTS OF TEACHING, IN ORDER TO CHECK THE ABILITY TO ANALYZING AND PRESENTING WITH RIGOR THE PROPERTIES OF NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS PRESENTED DURING THE LESSONS.
Texts
J.D.LAMBERT, NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL SYSTEMS, J. WILEY & SONS, 1991.
SLIDES ON THE TEAMS WEBSITE