MM546: Partial differential equations: theory, modelling and simulation
Study Board of Science
Teaching language: Danish or English depending on the teacher, but English if international students are enrolled
EKA: N300008102
Assessment: Second examiner: Internal
Grading: 7-point grading scale
Offered in: Odense
Offered in: Autumn
Level: Bachelor
STADS ID (UVA): N300008101
ECTS value: 10
Date of Approval: 25-04-2019
Duration: 1 semester
Version: Archive
Comment
Entry requirements
Academic preconditions
Students taking the course are expected to:
- Have knowledge of calculus, linear algebra, real analysis, integration theory and Banachspaces.
- Be able to use python scripting and numerical methods to solve algebraic and ordinary differential equations.
Course introduction
The aim of the course is 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.
engineering by partial differential equations. To analyze and solve
these equations both by analytic tools (when appropriate) and by
computational methods.
The course builds on the knowledge acquired in
the courses MM536 (Calculus for mathematics), MM538 (Algebra and linear
algebra), MM533 (Mathematical and Numerical Analysis), MM547 (Ordinary
differential equations: theory, modelling and simulation), and MM548
(measure- and integration theory, Banachspaces).
The course is of
high multidisciplinary value and gives an academic basis for a Bachelor
Project in several core areas of Natural Sciences.
high multidisciplinary value and gives an academic basis for a Bachelor
Project in several core areas of Natural Sciences.
In relation to the competence profile of the degree it is the explicit focus of the course to:
- Give the competence to :
- handle complex and development-oriented situations in study and work contexts.
- Identify needs and plan individual learning
- Give skills to:
- analyze practical and theoretical problems using numerical simulations based on suitable mathematical models.
- Analyze qualitative and quantitative properties of mathematical models.
- Explain and evaluate errors in modeling and simulation
- Explain and select relevant analytic and solution models
- Formulate and present problems and solutions to fellow students and partners
- Give knowledge and understanding of:
- mathematical modelling and numerical analysis of problems in natural science and engineering.
- theory, methods and praxis within applied mathematics.
Expected learning outcome
The learning objectives of the course are that the student demonstrates the ability to:
- Formulate a partial differential equation as a model for a simple problem.
- Classify 2nd order PDEs and describe their characteristic properties.
- Analyze and simulate partial differential equations by the methods taught in the course.
- Construct, implement and analyze numerical methods to compute (approximate) solutions to partial differential equations.
- Give a seminar presentation of the individual project and answer supplementary questions.
Content
The following main topics are contained in the course:
- Classification of 2nd order PDEs: Elliptic, parabolic and hyperbolic problems.
- Elliptic boundary value problems and Galerkin Finite Elements.
- Variational formulation, ellipticity, and the Lax-Milgram theorem.
- Sobolev spaces, Cauchy-Schwarz and Poincare inequalities.
- The poisson equation: Variational form, ellipticity, and FEniCS implementation.
- Galerkin's method, Galerkin orthogonality, best approximation, and error analysis.
- Finite elements for the Poisson equation, error bounds by duality.
- Neumann, Dirichlet, and Robin boundary conditions.
- div-grad operators and FEniCS.
- Parabolic PDEs: The heat equation.
- Runge-Kutta time stepping in variational form.
- SDIRK methods and L-stability.
- Simulation of the heat transfer.
- Parabolic-elliptic systems: Navier-Stokes equations
- Chorin’s projection method.
- Incremental pressure correction – IPC method
- Simulation of incompressible flow with heat transfer
- Adaptive calibration of PDE models
Literature
Examination regulations
Exam element a)
Timing
Autumn
Tests
Project assignment with oral presentation
EKA
N300008102
Assessment
Second examiner: Internal
Grading
7-point grading scale
Identification
Student Identification Card
Language
Normally, the same as teaching language
Examination aids
To be announced during the course
ECTS value
10
Additional information
Reexam in the same exam period or immediately thereafter.
The reexam may be a different type than the ordinary exam.