Finite Volume Methods
Finita volymmetoder

FMN015F, 7.5 credits

Valid from: Autumn 2014
Decided by: FN1/Anders Gustafsson
Date of establishment: 2015-03-15

General Information

Division: Numerical Analysis
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: NUMN14
Teaching language: English

Aim

Finite Volume methods are the standard numerical methods for the solution of conservation laws, which represent fundamental laws of physics. Of particular importance is the use of the latter to model fluid flows in the form of parabolic and hyperbolic partial differential equations. The course explains basic pitfalls of numerical methods for these equations and how to arrive at stable and convergent finite volume methods of first order. The course is necessary for further studies within numerical analysis and also useful for students within applied disciplins where conservation laws are used.

Goals

Knowledge and Understanding

For a passing grade the doctoral student must demonstrate deep knowledge of mathematical and numerical difficulties regarding shock waves.

Competences and Skills

For a passing grade the doctoral student must

• be able to independently choose, implement and use advanced computational methods for conservation laws.
• be able to report solutions and numerical simulations in written form.
• be able to judge the accuracy and relevance of numerical results

Course Contents

Conservation Laws, Reynolds' Transport theorem, Navier-Stokes equations Upwind methods and central discretizations Stability and the Courant-Friedrichs-Lewys (CFL) condition The theorem of Lax Wendroff Characteristics, Linear Systems Nonlinear systems, Roe's method Uniqueness, Entropy solutions, Entropy condition Finite Volume methods in multiple dimensions Boundary conditions Time Integration Higher Order, Theorem of Godunov, Discontinuous Galerkin (DG) Methods Quadrature, DG-Spectral Element Methods Stability of DG methods, Time Integration aspects

Course Literature

• Numerical Methods for Conservartion Laws.. ISBN 3764324643.
• Birken, P.: Numerical Methods for the Unsteady Navier-Stokes equations. Habilitation Thesis, University of Kassel.. 2012.

The course participants may download the second text from the course homepage.

Instruction Details

Types of instruction: Lectures, exercises

Examination Details

Examination formats: Oral exam, written assignments
Grading scale: Failed, pass
Examiner:

Admission Details

Assumed prior knowledge: Vector analysis. Programming in Python or Matlab.

Further Information

Contact person: Philipp Birken, philipp.birken@na.lu.se

Course Occasion Information

Contact and Other Information

Course coordinators:
Web page: http://www.ctr.maths.lu.se/course/finitvol/