# MA8103 Nonlinear Partial Differential Equations

### 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 Email: holden [at] math [dot] ntnu [dot] no

**Textbook:** H. Holden and N. H. Risebro: Front Tracking for Hyperbolic Conservation Laws, Springer, Corr. 2nd printing 2007. The book exists as an eBook, and NTNU students can read and download it freely from the website eBook. One can also buy it directly from that web site for EUR 24.95.

### Time and place

Tuesday at 14:15-16 in room 734, Sentralbygg 2 Friday at 8:15-10 in room 734, Sentralbygg 2

The lectures will be in English, and there will be an oral exam at the end of the semester.

### Lecture plan

- Fri, Jan. 10: p. 1-8
- Tue, Jan. 14: p. 9-11, example with linear model, convergence of upwind scheme for the linear advection equation. MATLAB characteristics program
- Fri, Jan. 17: Entropy condition for the linear advection equation
- Tue, Jan. 21: Precise definition of weak entropy solution. Pages 23-26 (Ch. 2).
- Fri, Jan. 24: p. 26-32 Linear equations
- Tue, Jan. 28: p. 32-37
- Fri, Jan. 31: p. 37-43
- Tue, Feb. 4: p. 44-49
- Fri, Feb. 7: p. 50-53
- Tue, Feb. 11: p. 53-55
- Fri, Feb. 14: p. 63-68
- Tue, Feb. 18: p. 69-72
- Fri, Feb. 21: p. 73-76
- Tue, Feb. 25: p. 76-79
- Fri, Feb. 28: p. 79-81, Higher-order methods
- Tue, March 4: p. 117-119
- Fri, March 7: p. 120-125
- Tue, March 11: p. 125-129
- Fri, March 14: p. 129-133
- Tue, March 18: p. 39-45 in Linear equations
- Fri, March 21: p. 45-46 in Linear equations and pp. 163-170
- Tue, March 25: p. 170-178
- Fri, March 28: p. 178-183
- Tue, April 1: p. 183-185
- Fri, April 4: p. 185-189
- Tue, April 8: p. 189-196
- Fri, April 11: p. 196-197.

### Recommend exercises

- Ch. 1: Exercises 1.2, 1.3, 1.6, 1.7
- Ch. 2: Exercises 2.1, 2.9, 2.13-17
- Ch. 3: Exercises 3.1, 3.2,

### Additional material

- characteristics.m: MATLAB characteristics program
- convex_envelope.zip: MATLAB program computing the convex envelope of a piecewise linear approximation of a function. Extract to an empty folder and run test.m for an example.