MM530: Partial Differential Equations and Numerics (10 ECTS)

STADS: 13010101

Level
Bachelor course

Teaching period
The course is offered in the autumn semester.

Teacher responsible
Email: Achim@imada.sdu.dk

Timetable
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Comment:
Ubegrænset deltagerantal. Fælles undervisning med MM546 og MM834

Prerequisites:
None

Academic preconditions:
Linear Algebra, Mathematical and Numerical Analysis and Python scripting should be known.

Course introduction
To introduce modeling of problems from science and engineering by partial differential equations. To analyze and solve these equations both by analytic tools (when appropriate) and by computational methods.

Expected learning outcome

1. Formulate a partial differential equation as a model for a simple problem.
2. Classify 2nd order PDEs and describe their characteristic properties.
3. Analyze and simulate partial differential equations by the methods taught in the course.
4. Construct, implement and analyse numerical methods to compute (approximate) solutions to partial differential equations.
5. Give a seminar presentation and answer supplementary questions on the course syllabus and the problems solved in mandatory assignments. 

Subject overview

1. Boundary value problems, finite differences and the curse of condition.
2. Classification of 2nd order PDEs: Elliptic, parabolic and hyperbolic problems.
3. Elliptic boundary value problems and Galerkin Finite Elements.
1. Variational formulation, ellipticity, and the Lax-Milgram theorem.
2. Sobolev spaces, Cauchy-Schwarz and Poincare inequalities.
3. The poisson equation: Variational form, ellipticity, and FEniCS implementation.
4. Galerkin's method, Galerkin orthogonality, best approximation, and error analysis.
5. Finite elements for the Poisson equation, error bounds by duality.
6. Neumann, Dirichlet, and Robin boundary conditions.
7. div-grad operators and FEniCS.
4. Parabolic PDEs: The heat equation.
1. Runge-Kutta time stepping in variational form.
2. SDIRK methods and L-stability.
3. Continous vs discontinous Galerkin time stepping methods.
4. Simulation of the heat distribution on a cooling fin.
5. Hyperbolic PDEs: The wave equation.
1. The Lax-Friedrichs scheme.
2. Simulation of interference in acoustic waves.
3. The Kreiss-matrix-theorem and well-posedness of hyperbolic problems.
4. Nonlinear waves, breaking of waves and shock waves. 

Literature

Meddeles ved kursets start

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
Prerequisite test consisting of mandatory assignments. A positive evaluation of the mandatory assignments is a pre-requisite for the oral exam. (13010112)

Assessment and marking:

1. Oral exam. Danish 7 mark scale, internal examiner. The oral exam is in seminar form. Seminars take place during tutorials. Students present their assignments as a seminar talk. Participation in seminars and tutorials is mandatory. (10 ECTS) (13010102)

Reexam in the same exam period or immediately thereafter. The reexam may be a different type than the ordinary exam.

Expected working hours
The teaching method is based on three phase model.
Intro phase: 56 hours
Skills training phase: 28 hours, hereof:
 - Tutorials: 14 hours
 - Laboratory exercises: 14 hours

Educational activities Study phase: 14 hours

Language
This course is taught in Danish or English, depending on the lecturer. However, if international students participate, the teaching language will always be English.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.