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Numerical Solution of Partial Differential Equations with the use of Compatible or 'Mimetic' Methods (2016)

ECTS credits: 5

 

Course parameters:
Language: English
Level of course: PhD course
Time of year:Q3 2016
No. of contact hours/hours in total incl. preparation, assignment(s) or the like:37/?

 

Objectives of the course:
Most numerical methods for solution of pdes are based on nodal solution of the unknowns. Here we want to introduce compatible or mimetic discretization methods, which mimic the continuous differential equations as much as possible. Our primary interest is not to minimize the truncation error or the residual – used in finite difference, finite element, and finite volume methods – instead we will use the geometric content of the physical variables to obtain compatible discretization schemes. This involves use of differential geometry instead of traditional vector calculus, which is inadequate to associate field variables to its geometric objects. What we obtain is that the discrete representation of the pde will conserve the property of the continous pde, being for example mass, momentum, energy, enstrophy, or helicity. The main theme throughout the course will be the solution of the incompressible, Navier-Stokes equation.

With the need for better, faster, and more accurate solution of pdes we want to introduce students to new techniques, which will have a profound effect on tomorrows engineering applications. It is highly relevant for anybody solving pdes. The students are required to have an intermediate knowledge of vector calculus and will be introduced to differential forms, exterior calculus, and algebraic topology.


Learning outcomes and competences:
At the end of the course, the student should be able to understand and use:

  • differential forms
  • exterior calculus
  • algebraic topology

Compulsory programme: 
To pass the course the students have to hand in two assignments at most 3 weeks after the course for final approval.

 

Course contents: 
See “Objectives of the course” and “Learning outcomes”.

 

Prerequisites:
Calculus, Fluid Dynamics

 

Name of lecturers: 
Marc Gerritsma, Prof. (Delft University) and Bo Gervang, Assoc. Prof. (Aarhus University)

 

Type of course/teaching methods: 
Lectures, exercises

 

Literature: 
Notes

 

Course homepage: 
None

 

Course assessment: 
Passed/not passed based on approval of assignments.

 

Provider:
Aarhus University, Department of Engineering, Mechanical and Materials Engineering

 

Special comments on this course:
None 

Time: 
14th of March to the 18th of March 2016, lectures start at 9 am and end at 4 pm.

 

Place: 
Navitas, Inge Lehmanns Gade 10, Aarhus University, 8000 C, Aarhus.

 

Registration:
Deadline for registration is Monday, 7 March 2016. Information regarding admission will be sent out no later than Wenesday, 9 March 2016.

For registration: Please contact Bo Gervang, e-mail: bge@ase.au.dk

Comments on content: 
Revised 20.06.2016