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NUMERICAL SOLUTIONS OF PDE'S A

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2021/2022
Year: 
1
Academic year in which the course will be held: 
2021/2022
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

The course is addressed to Mathematics students, but also to students of other degree programs, with interests in scientific computing. Basic notions of Analysis I and II ans Numerical Analysis are required.
For the practical part, a basic knowledge of a programming language is needed. We will use mostly MatLab, but each student will be free to employ other ones, like C, C++ or SciPy.

Final Examination: 
Orale

The exam is oral, and consists of two parts, which take place on the same day.
In the first part, the student discusses a computational project, agreed with the teacher and submitted together with the MatLab code developed. The project shall be a case-study that leverages on a topic taught during the course. The subject of the evaluation will be the appropriateness and quality of the software produced and the ability to present and discuss critically the results.
The second part of the exam is an oral examination of the material covered by the course. The evaluation will be based on knowledge of the course contents, ability to master the technical jargon of the subject, critical reasoning and ability to link the various topics.

Assessment: 
Voto Finale

The two courses Numerical Solutions of PDEs (A and B) introduce the students to the numerical techniques to approximate the solutions of partial differential equations, which are at the foundation of many mathematical models. In particular the A course is focused on the finite volume approach and its application in the context of system of nonlinear conservation laws.

Conservation laws are present in most physical models, since they represent mathematically phisical laws like the conservation of mass, of momentum, of energy, etc. Moreover they allow to formulate also less classical models like those for vehicular traffic. In particular the LWR model of traffic, the Saint-Venant (shallow water) and the Euler gasdynamics equations will be treated explicitely in the course.

At the end of the course the student should be able to solve numerically systems of nonlinear conservation laws with the finite volume method. Moreover, he/she should be able to use critically also libraries and software based on finite volumes.

1.Conservation and balance laws. Strong and weak form of the equations. Method of characteristics, rarefactions and Rankine-Hugoniot solutions, entropy.
2. Linear methods for linear equations: exmaples, consistency, stability, CFL condition, Von Neumann analysis.
3. Conservative methods; lax-Wendroff Theorem; Godunov method. Solution of Riemann problems for scalar equations and associated Godunov method.
4. Shocks and Hugoniot locus; rarefactions and integral curves; Riemann problem for the shallow water equations; Godunov method for systems.
5. Approximate Riemann solvers; Roe’s and HLL methods.
6. Nonlinear stability; convergence of the methods for nonlinear equations; methods with bounded total variation.
7. TVD methods with second order of accuracy and hints for higher order methods
8. Complements: balance laws, numerical entropy production.

The lectures will follow mainly:
1) LeVeque – Numerical Methods for conservation laws – Birkhauser
Fro some topics, these are also useful:
2) LeVeque – Finite Volume Methods for Hyperbolic Problems – Cambridge Univ. press
3) Toro – Riemann Solvers and Numerical Methods for Fluid Dynamics - Springer

Additional material, especially for the lab sessions, will be made available by the teacher through the e-learning web site.

Lectures (2/3 of the hours) are conducted mainly at the blackboard. Exercises to help the individual study of each pupil will be made available and discussed during the following lectures upon request.
One third of the hours will be in the computing lab to teach the students how to implement, test and employ algorithms based on finite elements (some of the tools explained in the lectures will be used for this).

Office hours are booked on demand, by email or at lecture time.

Professors

SEMPLICE MATTEO

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