MA8103 Non-Linear Hyperbolic Conservation Laws

Spring term 2024

Lecture times:

Mondays 13:15-15:00 in EL23

Wednesdays 10:15-12:00 in F404 (except in week 5, where we have to be in B23)

We start on Mon 22 Jan at 13:15.

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.

Lecturer: Helge Holden, Harald Hanche-Olsen

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 € 39.99 (incl. shipping). See the upper right on the linked page. (If you don't see it at that price, perhaps you need to be on the campus network, either physically or via VPN.)


Week Date pages Etc
4 Mon 22.01 Ch. 1, p. 1–9
Wed 24.01 Ch. 1, p. 9–12, Ch. 2, p. 53–55 Please do the rest of Example 1.6 and Exercises 1.8 and 1.9
5 Mon 29.01 (Harald lectured) Ch. 2, p. 55–59 Suggest you work through the bottom half of page 59 on your own.
Wed 31.01 Ch. 2, Sec. 2.2 (p. 60-66) In room B23!
6 Mon 05.02 Ch. 2, Sec. 2.3 (p. 66-73) We stated, but did not prove Corollary 2.8
Wed 07.02 Ch. 2, proof of Cor. 2.8, Sec. 2.4, p. 74-78
7 Mon 12.02 Ch. 2, Sec. 2.4, start with (2.61) We covered the rest of the proof of Prop. 2.10, and the case of convex fluxes for Lemma 2.11
Wed 14.02 We proved the rest of Lemma 2.11, and Thms. 2.14 and 2.15 We did not show the proof of Lemma 2.13
8 Mon 19.02 We proved the TVD property (Thm. 2.15) by first studying total variation (App. A, pp. 427-429). Then we started on Ch.3, covering pp. 95-98. We gave the definition of the Lax-Friedrichs scheme
Wed 21.02 (Harald lectures this day plus week 9)
Started on p. 99, defining the Godunov scheme. Covered consistency and monotonicity (for the latter, see bottom of p. 107) and argued that the Godunov and Lax–Friedrichs methods are both consistent and monotone. Finally, started on the local truncation error.
Skipped most of p. 100–103 (higher order, multistep methods).
9 Mon 26.02 p. 104–108, also Theorem 3.10 on p. 115, skipping its proof starting near the top of p. 112.
Wed 28.02 p. 109–110 [middle of the page), but with quite different proofs based on the Kolmogorov–Riesz compactness theorem and the Arzelá–Ascoli thoerem.
(Back to Helge next week.)
See our paper on Kolmogorov–Riesz (arXiv/published version)
Notes on total boundedness and Arzelà–Ascoli
10 Mon 04.03 Theorem 3.9 proved, Theorem 3.10 mentioned, but not proved. Ch. 4, p. 171-174.
Wed 06.03 Ch. 4, p. 174-180.
11 Mon 11.03 (Harald lectures until Easter)
Ch. 5 (The Riemann Problem for Systems), p. 223–235, except the \(\epsilon\)-parametrization starting at the bottom of p. 231 (we'll get to that) and the discussion of contact discontinuities (degenerate waves) on p. 232.
Wed 13.03 No lecture
12 Mon 18.03 Ch. 5, p. 232 (contact discontinuities), 235–240 except the Hugoniot locus for the shallow water equation. Major departure from the book: Get the local \(j\)-shock curve by solving \(u(\epsilon)-u_L=\epsilon v_j(u(\epsilon),u_L)\) for \(u(\epsilon)\) with the aid of the implicit function theorem. Note: Better approach for the local Hugoniot locus. I also included a few words on the smooth variation of eigenvalues and -vectors, since I was a but unclear in the lecture.
Wed 20.03 Ch. 5 (cont'd.): The Hugoniot locus for the shallow water equations, including energy/entropy considerations.
pp. 241–243, plus the parts from 235–240 skipped on Monday.
Notes: Rankine–Hugoniot for shallow water (new version uploaded 2024-03-22)
and Energy for shallow water (new)
13–14 Mon
Easter (no lecture)
14 Wed 03.04 Uncertainty quantification for conservation laws pp. 239–244 Uncertainty quantification for conservation laws
15 Mon 08.04 pp. 245–247, 256 (Lemma 5.20)–259
Wed 10.04 pp. 259–262
16 Mon 15.04 (Harald lectured.)
First, some repetition from before Easter.
Then a bit on viscous profiles for systems: See notes. Also the book pp. 244–248, but I skipped the detailed computation for the shallow water equations.
Next Lax's theorem pp. 254–256.
Finally on front tracking, see Ch 6: Just a description of the algorithm, no proofs. Ended with a brief description of Glimm's random choice method
Note on viscous profiles
Wed 17.04
17 Mon 22.04 Equivalence of the Eulerian and Lagrangian viewpoints.
This is not in the book.
Note on Eulerian vs Lagrangian
Wed 24.04 Case study: The Euler equations. See pp. 260–270 in the book, but my appraoach is different. See the notes. (I did not quite have enough time for transforming the static shocks to general shocks, but that is covered in the notes.) Note on the Saint-Venant and Euler equations
18 Mon 29.04 Sec. 4.5. Operator Splitting: Source


Curriculum: Ch. 1: p. 1–16. Ch. 2: all. Ch. 3: p. 95–111. Ch. 4: p. 171–180, 212-217. Ch. 5: p. 223–277. Ch. 6: p. 283–296 with little emphasis on proofs. In addition, notes on uncertainty quantification (except sections 5.1.2 and 5.2.3) and notes on the equivalence of Euler and Lagrange formulations.

Exam is oral. You begin with a 20 min presentation of one of the topics given below. The choice of topic is randomly chosen from the topics given below at the beginning of the exam. You may use any aids in this part of the exam. After that you put all your notes aside, and you answer questions from the faglærer from any part of the curriculum. For the whole exam you use chalk and blackboard only, in particular, no powerpoint presentation. Total duration of the exam is 45 min. The grade is pass/fail.

Topics: A: Ch. 2. B: Ch. 3 (p. 95–111). C: Ch. 4 (p. 171–180, 212-217). D: Ch. 5 (p. 223–277). E: uncertainty quantification (except sections 5.1.2 and 5.2.3). F: Sec. 5.6 (the Riemann problem for the Euler equations), and the equivalence of Euler and Lagrange formulations.

Advice: You will not be able cover that much material in 20 min. Please select a part of it where you can give more details. Please select something that is not too trivial, and remember to include at least one proof.

Time for exam (Venue to be decided)

Tue, June 4,,

Wed, June 12,,

2024-04-29, Harald Hanche-Olsen