NUMERICAL SOLUTIONS OF PDE'S B
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
The course is designed for students in Mathematics and Physics, but it is conceived also for students with a different background, but with an interest in scientific computing. The main tools needed from analysis are Taylor expansions, vector and function norms. From numerical analysis, notions on polynomial interpolation and numerical integration of ordinary differential equations are useful, but they will be recalled.
As a programming language we will use Matlab, but each student is free to use other languages, such as C, C ++ or Fortran.
The exam is an oral exam, and consists of two parts, which can also be held on the same day.
In the first part, the student presents a brief report, which can also be carried out by groups of 2 or 3 people, on a project related to one of the course topics. Usually the project requires some programming, and the student must demonstrate a certain level of autonomy.
The second part of the exam is instead an oral exam on the topics covered during the course.
The course on Numerical methods for PDE's, part B concentrates on the numerical integration of hyperbolic systems of differential equations. This kind of equations arises when signals travel with finite speed. The most classic application is gas dynamics, but models for vehicular traffic flows, the evolution of stars in astrophysics, water flow with a free surface or plasma physics are all represented by equations with a hyperbolic structure.
Usually, hyperbolic conservation laws are not treated in classical courses in analysis, and therefore the course will be dedicated to an introduction to the solution of hyperbolic equations, both from an analytic and a numerical point of view. The study will concentrate on the qualitative behavior of solutions rather than on the analysis of the equations. The numerical schemes introduced will be finite volume methods, which are the standard schemes for these applications
During the course, we will study hyperbolic equations and some numerical methods developed to approximate them. The class will be structured by lectures with laboratory sessions, in which the behavior of the equations and of approximating methods will be observed.
At the end of the course, students should be able to resolve numerically hyperbolic PDEs on simple domains. Above all, they should be able to use software written by others in a critical and informed way.
- Scalar conservation laws. Characteristics, shock formation and the entropy condition. Weak solutions.
- Traffic models.
- Numerical methods for scalar equations. The linear case: Lax’ equivalence theorem. The nonlinear case: conservative methods and Lax Wendroff ‘s theorem. Solution of Riemann problems and Godunov’s method.
- Hyperbolic systems. Solution of the Riemann problem in the linear case. Non-linear case: rarefaction and shock waves. An example: the shallow water model and the resolution of the Riemann problem.
- Balance laws and well-balanced schemes.
- R. Leveque, “Finite volume methods for hyperbolic problems”, Cambridge.
- Notes and slides from the teacher
Lectures are traditional blackboard lectures, with practical sessions at the computer lab. In the lessons, we will introduce and gradually describe the numerical methods, together with the underlying theory. In the lab sessions, we will apply the methods studied to significant test cases, using programs written on purpose for this class. The aim is to understand by direct experience the structure of solutions to conservation laws and the characteristics of finite volume schemes.
Office hours are by appointment, which can be set either by email or at the end of the lessons
Professors
Borrowers
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Degree course in:
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Degree course in: PHYSICS