MM830: Partial Differential Equations and Numerics (10 ECTS)

STADS: 13010501

Level
Master's level course

Teaching period
The course is offered in the autumn semester.

Teacher responsible
Email: achim@imada.sdu.dk

Timetable
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Comment:
Samlæses med MM530

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. Be able to deal with partial differential equation models for complex processes in science.
2. Classify 2nd order PDEs and describe their characteristic properties.
3. Analyze and simulate partial differential equations using appropriate, advanced methods and modern software.
4. Design and perform reliable simulations of PDE models for complex processes in science.
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
There isn't any litterature for the course at the moment.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
Prerequisite test consisting of mandatory assignments. (13010512)

Assessment and marking:
Oral exam, Danish 7 mark scale, internal examiner (10 ECTS). (13010502)

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

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.

Remarks
This course is taught together with MM530 Partial Differential Equations and Numerics. The course MM830 cannot be taken in addition to MM530.

Course enrollment
See deadline of enrolment.

Tuition fees for single courses
See fees for single courses.