Studying at the University of Verona
Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.
Study Plan
This information is intended exclusively for future freshmen who will enroll for the 2025/2026 academic year.If you are already enrolled in this course of study, consult the information available on the course page:
Laurea magistrale in Mathematics [LM-40] - Enrollment until 2024/2025The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.
Please select your Study Plan based on your enrollment year.
1° Year
| Modules | Credits | TAF | SSD |
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2° Year It will be activated in the A.Y. 2026/2027
| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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2 modules among the following: area algebra and geometry + analysis
-A.A. 2025/2026 Homological Algebra not activated.
3 modules among the following: area modeling and computational mathematics24 credits among the following courses
- A.A. 2025/2026 Homological Algebra not activated.
- A.A. 2025/2026 Physics education laboratory not activated.Legend | Type of training activity (TTA)
TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.
Numerical methods for science and engineering: hyperbolic pdes and parallel computing (2025/2026)
Teaching code
4S013638
Teacher
Coordinator
Credits
6
Also offered in courses:
- Data Fitting and Reconstruction of the course Master's degree in Mathematics
Language
English
Scientific Disciplinary Sector (SSD)
MAT/08 - NUMERICAL ANALYSIS
Period
1st semester dal Oct 1, 2025 al Jan 30, 2026.
Courses Single
Authorized
Learning objectives
The course will discuss the theoretical and practical aspects and the possible applications of the numerical methods for the solution of partial differential equations of hyperbolic type.
We will focus in particular on Finite volume and discontinuous Galerkin methods with emphasis on the high order of accuracy. An integral part of the course will be the laboratory in which the methods presented during the lectures will be implemented in a programming language suitable for scientific computing. Basic knowledge of parallel computing will also be provided and it will be shown how to parallelise some of the algorithms studied in the course. At the end of the course, students are expected to have knowledge and skills in numerical analysis concerning hyperbolic equations and the numerical methods for their solution, to be able to evaluate both their limitations and potential, and to show excellent competence on their implementation.
Prerequisites and basic notions
Students must have completed a bachelor's degree in mathematics, computer science or engineering.
In addition, they must possess knowledge and skills in linear algebra, differential calculus in one and several variables, integral calculus, fundaments of differential equations, and the main methods of numerical calculus.
Program
- Introduction and revision of basical concepts
- Linear & nonlinear hyperbolic equations and systems of hyperbolic equations (e.g. LAE, Burgers, Shallow Water, Euler)
- Numerical methods for linear advection equations and their analysis
- Monotone methods and Godunov theorem
- Finite Volume schemes
• The method in 1D
• Godunov method
• Properties
• Exact Riemann solvers
• Approximate Riemann Solvers
• Finite volume schemes on unstructured two-dimensional grids
- High order Finite Volume schemes
• Second order schemes
• TVD reconstruction and limiters
• High order ENO and WENO nonlinear reconstruction procedures
•Business/industrial case studies, also in collaboration with international partner locations
• High order reconstruction in time: the ADER approach
- Discontinuous Galerkin methods
- Arbitrary-Lagrangian-Eulerian methods
- Programming in a compiled language (Fortran)
- Elements of parallel programming: the OpenMP paradigm
- Elements of parallel programming: the MPI paradigm
Depending on time and students preferences some of the following topics may be studied:
- Delaunay triangulations & Voronoi diagrams
- Path-conservative FV schemes
- Semi-implicit schemes for the incompressible Navier-Stokes equations
Bibliography
Didactic methods
Classroom theory lectures:
i) explanations will be given using the blackboard (chalkboard if available, or otherwise a projected graphics tablet);
ii) the lectures will focus on the theoretical explanation of the topics and the demonstration/justification of their properties;
iii) the reasons for the importance of the topics covered will also be provided, along with links and comparisons between related topics, comments on efficiency and possibilities of use;
iv) possible applications will be explained.
Numerous laboratory lectures will be dedicated to the implementation, motivation, and discussion of the numerical methods covered in the course. For this part, a laptop with MATLAB and a Fortran compiler installed is required.
It is important to note that the course aims to develop a mindset capable of inventing, developing, mathematically motivating, structuring, and effectively implementing new and effective algorithms for solving hyperbolic equations. To this end, the course starts with the theoretical study, motivations, and, above all, the constructive implementation of known algorithms. The use of artificial intelligence for their implementation is therefore NOT permitted because it would prevent students from learning the mechanism underlying the creation of algorithms and codes, completely replacing it in the case of simple algorithms, which, however, represent the tool for learning.
Learning assessment procedures
The exam consists of (A) a written test on theory and computational exercises to be completed within a set time limit and (B) an oral exam.
During the oral exam (B), you will be asked to i) present and comment on some of the methods implemented in class, ii) present and comment on one of the exercises assigned as homework, iii) answer theory questions in a structured manner, demonstrating your ability to make connections, and iv) demonstrate your skills.
You are only admitted to test (B) if you obtain a grade greater than or equal to 18/30 in test (A).
Evaluation criteria
It will be verified that the student
- knows the theoretical concepts covered in the course and is able to establish relationships and comparisons between them;
- demonstrates knowledge and skills in the field of numerical analysis for hyperbolic equations, numerical methods for their solution, and serial and parallel programming;
- knows how to apply the knowledge acquired to both theoretical and implementation exercises;
- can clearly explain concepts in written and oral form.
- can implement algorithms from scratch, modify and/or use existing algorithms.
- has developed a mindset suited to creating, developing, and implementing algorithms and codes.
Criteria for the composition of the final grade
Test A is considered passed when a grade of 18 out of 30 or higher is obtained.
Test B can only be taken once Test A has been passed.
The final grade (between 18 and 30L) is obtained by calculating the weighted average of the grades for Test A (40%) and Test B (60%).
One point will be added to the final grade if the student has attended at least 75% of the lessons in person.
Exam language
Inglese o Italiano
