# EP8410 - Numerical Methods for Hyperbolic Problems in Fluid Dynamics

### Examination arrangement

Examination arrangement: Written examination

Evaluation form Weighting Duration Examination aids Grade deviation
Written examination 100/100 4 hours D

### Examination arrangement

Examination arrangement: Oral examination

Evaluation form Weighting Duration Examination aids Grade deviation
Oral examination 100/100 D

### Course content

The course is taught every second year, next time will be autumn 2017. The PhD course will give an overview over numerical methods for hyperbolic problems in fluid dynamics. Hyperbolic partial differential equations govern waves and advection in fluid dynamics. We shall consider prominent examples like the Euler equations in gas dynamics, the acoustic wave equation in hydro- and aeroacoustics, the shallow water equations in hydraulics, and the drift flux equations in multiphase flow. Nonlinearities in hyperbolic equations and discontinuities, e.g. shocks, in their solutions are challenges.

We shall use the mathematical theory of hyperbolic systems and of nonlinear conservation laws to derive numerical methods and boundary conditions for hyperbolic problems.

We shall focus on finite volume methods, finite difference methods and discontinuous Galerkin methods for nonlinear scalar and system conservation laws in one and multiple dimensions. Godunov's method and approximate Riemann solvers will be presented. Total variation diminishing (TVD), essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) methods will be employed to compute flows with shocks and contact discontinuities.

### Learning outcome

The participants will get an overview over numerical methods for hyperbolic problems in fluid dynamics, insight into the use of mathematical theory for hyperbolic systems and nonlinear conservation laws to derive numerical methods and boundary conditions as well as training in the application of numerical methods to solve hyperbolic partial differential equations in fluid dynamics.

Knowledge: - After completion of this course, the student will have knowledge on:  Linear advection equation.  Linearized Euler equations.  Inviscid Burgers equation.  Euler equations.  Acoustic wave equation.  Shallow water equations.  Drift flux model.  Two-fluid model.  Linear and nonlinear hyperbolic systems.  Conservation laws.  Riemann problem.  Finite difference methods.  Finite volume methods.  Total variation diminishing (TVD) methods.  Godunov method.  Approximate Riemann solvers.  Essentially non-oscillatory (ENO) methods.  Weighted essentially non-oscillatory (WENO) methods.  Discontinuous Galerkin methods.

Skills: - After completion of this course, the student will have skills on:  Practical use and programming of numerical methods for hyperbolic problems in fluid dynamics.  Mathematical analysis of linear and nonlinear hyperbolic systems.  Derivation and implementation of characteristic boundary conditions.  Discretization of hyperbolic conservation laws with finite volume, difference and element methods.  Consistency analysis and von Neumann stability analysis for numerical methods for hyperbolic problems.  Computation of hyperbolic problems with shocks.  Checking and assessing the accuracy of numerical results for hyperbolic problems.

General competence: - After completion of this course, the student will have general competence on:  Numerical solution of hyperbolic fluid flow problems with finite volume, difference and element methods.  Mathematical analysis of hyperbolic systems.  Analysis of numerical methods for hyperbolic problems.

### Learning methods and activities

Lectures and written exercises. The teaching will be in English when students who do not speak Norwegian take the course. If the teaching is given in English the examination papers will be given in English only. Students are free to choose Norwegian or English for written assessments.
If there is a re-sit examination, the examination form may be changed from written to oral.
To pass the course a score of at least 70 percent is required.

### Required previous knowledge

Course TEP4165 Computational Heat and Fluid Flow or an equivalent CFD course.

### Course materials

Randall J. LeVeque: "Finite Volume Methods for Hyperbolic Problems." Cambridge University Press, Cambridge, 2002.

# Timetable

Detailed timetable

# Examination

Examination arrangement: Written examination

Term Statuskode Evaluation form Weighting Examination aids Date Time Room *
Autumn ORD Written examination 100/100 D 2017-12-06 09:00 E5

Examination arrangement: Oral examination

Term Statuskode Evaluation form Weighting Examination aids Date Time Room *
Spring ORD Oral examination 100/100 D 2018-05-30
• * The location (room) for a written examination is published 3 days before examination date.
If more than one room is listed, you will find your room at Studentweb.