Numeriska metoder för differentialekvationer

Valid for: 2024/25

Faculty: Faculty of Engineering LTH

Decided by: PLED F/Pi

Date of Decision: 2024-04-15

Effective: 2024-05-08

Main field: Technology
Depth of study relative to the degree requirements: Second cycle, in-depth level of the course cannot be classified

Mandatory for: F3, Pi3

Elective for: BME4, I4

Language of instruction: The course will be given in English on demand

The aim of the course is to teach computational methods for solving both ordinary and partial differential equations. This includes the construction, application and analysis of basic computational algorithms for approximate solution on a computer of initial value, boundary value and eigenvalue problems for ordinary differential equations, and for partial differential equations in one space and one time dimension. Independent problem solving using computers is a central part of the course.

Knowledge and understanding

For a passing grade the student must

- be able to describe the construction of basic numerical methods and algorithms in a logical and well structured manner, using suitable terminology
- be able to discretize ordinary and partial differential equations using finite difference and finite element methods, and to be able to independently implement and apply such algorithms
- be able to independently proceed from observation and interpretation of results to conclusion, and be able to present and account for his or her conclusions on a scientific basis in free report format.

Competences and skills

For a passing grade the student must

- be able to independently, on a scientific basis, select suitable computational algorithms for given problems
- be able to apply such computational algorithms to problems from applications
- be able to independently evaluate the relevance and accuracy of computational results
- present solutions of problems and numerical results in written form.

Judgement and approach

For a passing grade the student must

- be able to independently evaluate obtained numerical results in relation to the (unknown) solution of the differential equation studied
- be able to independently present results and conclusions of scientifically performed numerical experiments, in written or oral form, with references and other documentation of work carried out in support of their conclusions.

Methods for time integration: Euler’s method, the trapezoidal rule. Multistep methods: Adams' methods, backward differentiation formulae. Explicit and implicit Runge-Kutta methods. Error analysis, stability and convergence. Stiff problems and A-stability. Error control and adaptivity. The Poisson equation: Finite differences and the finite element method. Elliptic, parabolic and hyperbolic problems. Time dependent PDEs: Numerical schemes for the diffusion equation. Introduction to difference methods for conservation laws.

Grading scale: TH - (U, 3, 4, 5) - (Fail, Three, Four, Five)

Assessment: Written examination and computer projects. The final grade is the grade obtained on the written exam.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Modules

Code: 0122. Name: Written Examination.

Credits: 6.0. Grading scale: TH - (U, 3, 4, 5).
Assessment: Written examination.

Code: 0222. Name: Computer Projects.

Credits: 2.0. Grading scale: UG - (U, G).
Assessment: For each project, either a written or an oral presentation of the work. The specific form of presentation will be specified for each project at the start of the course.
The module includes: Three computational projects comprising implementation of numerical methods as computer programs, application of those on selected mathematical problems, and analysis of the results.

Assumed prior knowledge:
FMAB20 Linear Algebra, FMAB30 Calculus in Severable Variables, FMAN55 Applied Mathematics.

The number of participants is limited to: No

Kursen överlappar följande kurser:
NUMN12
NUMN20
NUMN32
FMN041
FMN050
FMN081
FMN130
FMNF10

- Iserles, A: Numerical analysis of differential equations. Cambridge University Press, 2008, ISBN: 978-0521734905.
- Lecture notes.

Course coordinator: Tony Stillfjord,
tony.stillfjord@math.lth.se

Director of studies: Studierektor Anders Holst,
Studierektor@math.lth.se

Course administrator: Student Office,
expedition@math.lth.se

Course homepage: https://canvas.education.lu.se/courses/20395