MA8103 Non-Linear 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 Hanche-Olsen 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
4	Fri	24 Jan	◐	pp. 55–59
5	Mon	27 Jan	◑	pp. 60–66
5	Fri	31 Jan		pp. 67–74
6	Mon	3 Feb		pp. 74–79
6	Fri	7 Feb	◐	pp. 80–83 (note)
7	Mon	10 Feb	◑	pp. 84–87. Recommended exercises: Ch.2: 1,3,10,16
7	Fri	14 Feb	◐	pp. 427–430 (note on BV · updated 2020-02-21), 95–97
8	Mon	17 Feb	◑	pp. 98–105
8	Fri	21 Feb	◐	pp. 105–109
9	Mon	24 Feb	◐	exercises; Kolmogorov–Riesz; pp. 109–110
9	Fri	28 Feb	◑	pp. 110–115; 171–173
10	Mon	2 Mar	◐	pp. 174–176
10	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.
11	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
13	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
14	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
15	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.
	Fri	17 Apr	◐	“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!