Nonlinear partial differential equations
The course will be given every other year.
Next time: Spring 2010.
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 Matematisk modellering 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.
In the course we will focus on fundamental properties of these equations and will discuss numerical methods for the computation of solutions. Knut-Andreas Lie has made a web site containing some examples of applications as well as some theory.
     There is extensive research activity in this area at the Department, and the course is well suited as a start of project, a master thesis or a PhD study. The course is useful for students from other departments who need to study numerical methods applicable to these equations.
     The course will be given by Helge Holden <>.
     The text book is Front Tracking for Hyperbolic Conservation Laws av H. Holden og N. H. Risebro.


The lectures will be in English.


From Holden og Risebro, Front Tracking for Hyperbolic Conservation Laws:
Ch. 1 All
Ch. 2 All
Ch. 3 Sect. 3.1
Ch. 4 Sects. 4.1 og 4.2.
Ch. 5 All
Ch. 6 Be able to explain the front-tracking methods for systems. No proofs.

Wed 13.1 p. 1-11  
Fri 15.1 p. 12-26  
Wed 20.1 p. 25-28  
Fri 22.1 p. 30-32  
Wed 27.1 p. 33-42  
Fri 29.1 p. 43-45  
Wed 3.2 p. 46-53  
Fri 5.2 p. 53-57  
Wed 10.2 p. 57-65  
Fri 12.2 p. 65-70  
Wed 17.2 p. 71-73  
Fri 19.2 p. 73-81 (The proof of Thm. 3.10 was not lectured)
Wed 24.2 p. 296-298, p. 117-120  
Fri 26.2 p. 121-124  
Wed 3.3 p. 125-128  
Fri 5.3 p. 128-133  
Mon 8.3 Exercise session Time: 13:15-14, room 922, Exercises: 2.1,2.15, 3.2
Wed 10.3 p. 163-166  
Fri 12.3 p. 167-171  
Wed 17.3 p. 172-182  
Fri 19.3 p. 183-186  
Wed 24.3 p. 187-188  
Fri 26.3 p. 189-191  
Wed 7.4 p. 192-194  
Fri 9.4 p. 195-200  
Wed 14.4 Exercise session Problems
Fri 16.4 Exercise session  
Wed 21.4    
Fri 23.4    


The exam is oral and individual. Each candidate is asked to present a 20 min lecture on an assigned topic (usually a chapter or a section from the text book). Please use blackboard (preferred) or overhead. Notes are allowed for the presentation. Try to present the key results, some proofs. The presentation should not be "popular", but rather focus on the main mathematical results and techniques. After that the candidate will be asked questions for another 20-25 min from all of the curriculum (given above).

The exam for 2010 will take place on June 4 and June 28 (the exact time and order of the candidates will be decided on the day of the exam):
June 4
Halvor Lund (topic: Ch. 2)
Karl Yngve Lervåg (topic: Sec. 3.1)
Zerihun Kinfe Birhanu (topic: Sec. 4.1-2)
Amund Stenseth (topic: Ch. 5)
Shengjie Lu (topic: Ch. 2)
Jan Chylik (topic: Sec. 3.1)
Kjetil Andre Johannessen (topic: Sec 3.1)
June 28
Alexandre Morin (topic: Sec. 4.1-2)
Håkon Marthinsen (topic: Ch. 5)
Gunhild A Gjøvåg (topic: Ch. 2)

2010-05-06 10:26