MM834: Partial differential equations: theory, modelling and simulation (10 ECTS)

STADS: 13016001

Level
Master's level course approved as PhD 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 Partielle differentialligninger: teori, modellering og beregning

Prerequisites:
A Bachelor’s degree in mathematics, physics or computer science.

The course cannot be chosen by students, who passed MM546.

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.

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 Master 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 :
1. handle complex and development-oriented situations in study and work contexts.
2. Identify needs and plan individual learning
• Give skills to:
1. analyze practical and theoretical problems using numerical simulations based on suitable mathematical models.
2. Analyze qualitative and quantitative properties of mathematical models.
3. Explain and evaluate errors in modeling and simulation
4. Explain and select relevant analytic and solution models
5. Formulate and present problems and solutions to fellow students and partners
• Give knowledge and understanding of:
1. mathematical modelling and numerical analysis of problems in natural science and engineering.
2. theory, methods and praxis within applied mathematics.

Expected learning outcome
The learning objectives of the course are that the student demonstrates the ability to:

• Be able to deal with partial differential equation models for complex processes in science.
• Classify 2nd order PDEs and describe their characteristic properties.
• Analyze and simulate partial differential equations using appropriate, advanced methods and modern software.
• Construct, implement and analyze numerical methods to compute (approximate) solutions to partial differential equations.
• Understand the mathematical theory for numerical methods for PDEs.
• Design and perform reliable simulations of PDE models for complex processes in science.
• Give a seminar presentation of the individual project and answer supplementary questions.

Subject overview
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

Meddeles ved kursets start.

Website
This course uses e-learn (blackboard).

Prerequisites for participating in the exam
None

Assessment and marking:
Project assignment with oral presentation. Internal marking, Danish 7-mark scale.

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: 52 hours
Skills training phase: 28 hours, hereof:
 - Tutorials: 28 hours

Educational activities

Educational form
Activities during the study phase:

• preparation of exercises in study groups
• preparation of projects

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.