MA8103 NonLinear Hyperbolic Conservation Laws
Spring term 2020
General information
Background: In the course we study a class of nonlinear partial differential equation called hyperbolic conservation laws. These equations are fundamental in our understanding of continuum mechanical systems, and can be used to describe mass, momentum and enery conservation in mechanical systems. Examples of the use of conservation laws you may have seen in TMA4305 Partial differential equations and TMA4195 Mathematical modeling as well as in courses in physics and fluid mechanics. The equations share many properties that make numerical computations difficult. The equations may, for instance, develop singularities in finite time from smooth initial data. These equations have been extensively studied due to their importance in applications. Examples of applications include weather forecasting, flow of oil in a petroleum reservoir, waves breaking at a shore, and in gas dynamics.
Lecturers: Harald HancheOlsen and Helge Holden
Textbook: H. Holden and N. H. Risebro: Front Tracking for Hyperbolic Conservation Laws, Springer, Second edition 2015. The book exists as an eBook, and NTNU students can read and download it free of charge. You can purchase a paperback edition called MyCopy for € 24.95 (incl. shipping). See the upper right on the linked page.
Lecture plan
Week  Date  Who  Material covered  

◐  means: Harald lectures  
◑  means: Helge lectures  
3  Fri  17 Jan  ◑  pp. –8. 
4  Mon  20 Jan  pp. 9–15, pp. 53–55  
Fri  24 Jan  ◐  pp. 55–59  
5  Mon  27 Jan  ◑  pp. 60–66 
Fri  31 Jan  pp. 67–74  
6  Mon  3 Feb  pp. 74–79  
Fri  7 Feb  ◐  pp. 80–83 (note)  
7  Mon  10 Feb  ◑  pp. 84–87. Recommended exercises: Ch.2: 1,3,10,16 
Fri  14 Feb  ◐  pp. 427–430 (note on BV · updated 20200221), 95–97  
8  Mon  17 Feb  ◑  pp. 98–105 
Fri  21 Feb  ◐  pp. 105–109  
9  Mon  24 Feb  exercises; Kolmogorov–Riesz; pp. 109–110  
Fri  28 Feb  ◑  pp. 110–115; 171–173  
10  Mon  2 Mar  ◐  pp. 174–176 
Fri  6 Mar  pp. 177–185  
11  Mon  9 Mar  ◑  pp. 186–188, 223–229. We skip the derivation of the shallow water equations from Navier–Stokes. 
Fri  13 Mar  Due to the new coronavirus situation, we are now experimenting with digital solutions. pp. 229–231: part 1 – part 2 

12  Mon  16 Mar  ◐  pp. 231–238: “slides”; pencast: part 1 · part 2 · part 3 
Tue  17 Mar  pp. 238–240: “slides”; pencast: part 4 (improved resolution)  
Fri  20 Mar  ◑  pp. 241–248: part a; part b; part c; part d  
13  Mon  23 Mar  ◐  pp. 249–256 (except Lemma 5.18): “slides”; pencast: part 1 · part 2 
Fri  27 Mar  ◑  pp. 256–277: part a; part b; part c  
14  Mon  30 Mar  pp. 283–312 (no proofs): part a; part b  
Fri  3 Apr  ◐  Correspondence between Eulerian and Lagrangian formulations of conservation laws Note · “slides” · pencast: part 1 · part 2 · part 3 

15  Mon  6 Apr  ◯  Easter, no lectures 
Fri  10 Apr  
16  Mon  13 Apr  
Fri  17 Apr  ◐  Exercises: 3.1, 3.3, 3.10; 4.1; 5.8, 5.9, 5.10 Exercise 3.1 is wrong. To fix, assume \(u^0_j=[j\le0]\) instead of \(u^0_j=[j\ge0]\) (using the Iverson bracket) For Exercise 3.10, read this before starting on it. And for Exercise 5.9, read this first. 

“Slides” · pencasts: 3.1 · 3.3 · 3.10 · 4.1 (newest) · 5.8 · 5.9 (new) · 5.10 (newer)  
17  Mon  20 Apr  ◑  Sec. 4.5. pp. 212–217. lecture 
And that's it! 