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# Details for the Course Syllabus for Course FMN015F valid from Autumn 2014

General
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.
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
Stability of DG methods, Time Integration aspects
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
Judgement and Approach
• For a passing grade the doctoral student must
Types of Instruction
• Lectures
• Exercises
Examination Formats
• Oral exam
• Written assignments
• Failed, pass
Assumed Prior Knowledge
• Vector analysis. Programming in Python or Matlab.
Selection Criteria
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.
Further Information
• Contact person: Philipp Birken, philipp.birken@na.lu.se
Course code
• FMN015F