The purpose of the workshop is to introduce theory and numerics of the conservation laws to the participants. In the first week, students will be introduced to the notion of weak solution, various versions of entropy.

In the second week lectures focus on numerical methods for conservation laws and vanishing viscosity approximation to the conservation laws. Notions of consistency, stability and convergence of numerical schemes along with many concrete examples (linear and nonlinear) will be presented. Monotone, TVD, l 1 contraction schemes and the relation between them will be discussed. In the third week, one set of lectures will deal with the connection between Hamilton Jacobi theory and conservation laws with convex flux, whereas the other set of lectures will be on system of conservation laws.

**Speakers**

1. V. D. Sharma (IIT Bombay)

2. R Radha (Univ. of Hyderabad)

3. G.D. Veerappa Gowda (TIFR-CAM)

4. Shyam Ghoshal (TIFR-CAM)

5. S. Baskar (IIT Bombay)

6. A. Adimurthi (TIFR-CAM)

**Syllabus:**

Week 1:

- (Module-1) Stability theory for ordinary differential equations, Lyapunov energy functional, Poincar ́e’s theorem, one dimensional bifurcation theory, different types of bifurcations.
- (Module-2) Method of characteristics, weak solution, R-H condition, non-uniqueness of weak solution, entropy conditions, irreversibility, interaction

Week 2:

- (Module-3) Finite difference equations, consistency, stability of numerical schemes, convergence of numerical solutions, conservative schemes, Lax-Wendroff’s theorem, Lax-Friedrichs scheme,Godunov scheme, Engquist-Osher scheme, monotone schemes, Godunov’s theorem, TVD schemes, l 1 -contraction schemes, numerical entropy inequality.
- (Module-4) Vanishing viscosity approximation of conservation laws, uniform BV estimates, l ∞ estimates for the solution, uniqueness of an entropy solution.

Week 3:

- (Module-5) Hamilton-Jacobi equations, Lagendre transformation, Hopf-Lax formula, Lax-Oleinik formula, Structure theorem for bounded entropy solutions, conservation laws with discontinuous

flux, interface entropy condition, uniqueness of (A,B)-entropy solution, L 1 stability. - (Module-6) System of conservation laws, R-H condition, Lax-entropy condition, shock and rarefaction waves, entropy solutions of the Cauchy problem, continuous dependence of solutions, uniqueness

of entropy solutions for systems,

**Pre-requisites**: Knowledge in basic ODE, PDE, Functional Analysis, Measure Theory and Numerical Analysis.

**References**

- E. Godlewski, P.A. Raviart, Hyperbolic systems of conservation laws, Collection Math ́ematiques et Applications de la SMAI, Ellipses, Paris (1991)
- E. Godlewski, P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws , Springer, New York (1996).
- H. Holden, N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Applied Mathematical Sciences, Springer, 2015.
- R.J. Leveque, Numerical methods for conservation laws, 2nd edition, Birkh ̈auser Basel, 2005.
- B. Perthame, Kinetic formulation of conservation laws, Oxford lecture series in Mathematics and its applications-21, Oxford university press, 2002.
- J. Smoller, Shock Waves and Reaction–Diffusion Equations, Springer, Berlin, 1994.
- G.B Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs, and Tracts, New York, 1999.

**Tentative Time Table**

Date |
9:30 to 11:00 |
11:30 to 13:00 |
14:00 to 15:00 |
15:30 to 16:30 |

4-06-2018 | VDS | RR | VDS & Tutor | RR & Tutor |

5-06-2018 | VDS | RR | VDS & Tutor | RR & Tutor |

6-06-2018 | VDS | RR | VDS & Tutor | RR & Tutor |

7-06-2018 | VDS | RR | VDS & Tutor | RR & Tutor |

8-06-2018 | VDS | RR | VDS & Tutor | RR & Tutor |

9-06-2018 | VDS | RR | VDS & Tutor | RR & Tutor |

11-06-2018 | GDV | SG | GDV & Tutor | SG & Tutor |

12-06-2018 | GDV | SG | GDV & Tutor | SG & Tutor |

13-06-2018 | GDV | SG | GDV & Tutor | SG & Tutor |

14-06-2018 | GDV | SG | GDV & Tutor | SG & Tutor |

15-06-2018 | GDV | SG | GDV & Tutor | SG & Tutor |

16-06-2018 | GDV | SG | GDV & Tutor | SG & Tutor |

18-06-2018 | SB | AA | SB & Tutor | AA & Tutor |

19-06-2018 | SB | AA | SB & Tutor | AA & Tutor |

20-06-2018 | SB | AA | SB & Tutor | AA & Tutor |

21-06-2018 | SB | AA | SB & Tutor | AA & Tutor |

22-06-2018 | SB | AA | SB & Tutor | AA & Tutor |

23-06-2018 | SB | AA | SB & Tutor | AA & Tutor |