Course program
1) Hyperbolic conservation laws
Theory.
Derivation of a hyperbolic conservation law, density, velocity, flux. Integral formulation and differential formulation. Review of the method of characteristics in the linear case, examples with source and reaction terms. Extension to the nonlinear case, first shock time. Solutions almost everywhere, weak solutions. Equivalence between the notions of classical and weak solutions for regular solutions. Rankine-Hugoniot condition. Riemann problem, existence of infinite weak solutions, shock curves and rarefaction waves. Entropic solutions, the vanishing viscosity method, entropy conditions, entropy and entropy flux pairs. Properties of the entropic solution. Generalizations: multi-dimensional scalar case; one-dimensional vector case, hyperbolic and strictly hyperbolic systems, wave equation; one-dimensional nonlinear systems, Euler's equations; multi-dimensional vector case. Physical speed and characteristic speed.
Numerical methods.
Nodal grid and barycentric grid, example of non-converging scheme to a weak solution. Schemes in conservative form, numerical flux, consistency and conservation of mass. Classical schemes (upwind, Lax-Friedrichs, Richtmyer-Lax-Wendroff, Mac Cormack), Godunov scheme, explicit Godunov flow in the cases f(u)=1/2u^2 and f(u)=u(1−u) , CFL condition. Lax-Wendroff theorem. Introduction to the discrete notion of entropy and entropy flux pairs. Overview of TVD schemes, monotonicity-preserving schemes and monotonic schemes, main results of convergence to the entropic solution.
Applications.
Vehicular traffic, first and second order models.
2) Hamilton-Jacobi equations
Theory.
Classical examples of Hamilton-Jacobi (HJ) equations, bridge theorem with hyperbolic conservation laws. Characteristic system for HJ equations, non uniqueness of weak solutions, viscosity solution and its properties. Legendre transform, representation formulas, Hopf-Lax formula. Optimal control problems, infinite horizon, finite horizon, minimum time. Value function, dynamic programming principle, Hamilton-Jacobi-Bellman (HJB) equation, optimal control in feedback form and optimal trajectories.
Numerical methods.
Numerical bridge theorem with hyperbolic conservation laws. Finite and semi-Lagrangian difference scheme for the evolutive eikonal equation. Finite difference scheme for the stationary eikonal equation. Semi-Lagrangian scheme for the stationary eikonal equation with proof of convergence.
Applications.
Numerical solution of the HJB equation using a semi-Lagrangian scheme. Overview on the front evolution problem and the level-set method. Brief description of some classic examples: the lunar landing; Zermelo's navigation problem; the stabilization of the inverted pendulum. Overview on some generalizations: state constraints, constraints on controls, hybrid dynamics, stochastic dynamics, differential games.
Prerequisites
The course of Institutions of Numerical Analysis is preparatory to this course.
Books
R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Basel.
L. C. Evans, Partial Differential Equations, AMS.
Lecture notes in pdf and additional material on the e-learning page of the course.
Frequency
Attendance is optional but strongly recommended given the way it is carried out.
Exam mode
Exam to-do list:
1. codes related to the schemes studied during the course.
2. written project on a chosen applicative topic and related code.
The exam therefore consists in the presentation of the chosen project and in an oral test on the topics covered in the course. The final vote will take into account all the elements indicated.
Bibliography
R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Basel.
L. C. Evans, Partial Differential Equations, AMS.
Lesson mode
The course takes place in the laboratory both for the theoretical part and for the part of exercises and code development.
