Numeriska simuleringar av flödesproblem

**Valid from:** Spring 2023**Decided by:** Maria Sandsten**Date of establishment:** 2022-12-22

**Division:** Numerical Analysis**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course codes:** NUMN28, FMNN40**Teaching language:** English

The overall goal of the course is that the graduate student should acquire basic knowledge in modern numerical methods for non-linear conservation laws, with a focus on fluid models. Important examples of such models are the Euler equations of gas dynamicsand the shallow water equations, both of which are simplifications of the Navier-Stokes equations. These models are used in the design of aircraft and wind turbines, as well as in climate system research. The course discusses so called finite volume methods for discretizing the models -- their derivation, convergence and stability properties, --- and touches upon higher order extensions. The discretization often leads to large nonlinear systems of equations. The course presents iterative methods for solving these -- such as Multigrid and Newton-Krylov. Their convergence properties are discussed, with a particular focus on systems arising from the above discretizations.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to give an account of mathematical and numerical difficulties arising with nonlinear conservation laws and shock solutions,
- be able to explain stability and convergence of discontinuous Galerkin methods,
- be able to describe the structure of Jacobian-free Newton-Krylov methods,
- be able to describe multi-grid methods and their use for flow problems.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to derive a discontinuous Galerkin method for a general conservation law,
- be able to implement a discontinuous Galerkin method for a one dimensional nonlinear conservation law,
- be able to interpret numerical stability and accuracy problems arising in simulations,
- be able to implement a Jacobian-free Newton-Krylov method with preconditioner,
- be able to implement a multigrid method and apply it to flow problems,
- be able to integrate knowledge from the various parts of the course to address problems within the framework of the course,
- be able to plan and execute qualified tasks within the framework of the course, with appropriate methods within given time-frames.

*Judgement and Approach*

For a passing grade the doctoral student must

- be able to critically evaluate and independently apply methods from the course within a project work,
- be able to evaluate their own responsibility for how the subject is
- used and discuss the subject's possibilities to contribute to a sustainable social development.

Models of computational fluid dynamics Hyperbolic conservation laws and their basic properties (weak solutions, weak entropy solutions, shocks) Discontinuous Galerkin discretizations Simulations of gas dynamics Krylov subspace methods with preconditioning Jacobian-free Newton-Krylov methods Multigrid methods for flow problems

- Birken, P.: Numerical Methods for Unsteady Compressible Flow Problems. CRC Press, 2021. ISBN 9780367457754.
- LeVeque, Randall J.: Numerical Methods for Conservation Laws. Springer Science & Business Media, 1992. ISBN 9783764327231.

First title also as Ebook.

**Types of instruction:** Lectures, project, miscellaneous.
Apart from lectures and a compulsory (final) project there are assignments. These are not compulsory but provides useful preparations for the project.

**Examination formats:** Oral exam, written report.
The examination consists of a written report of the project and an appurtenant oral examination based on the report. The oral examination is only given to those students who have passed the written report.**Grading scale:** Failed, pass**Examiner:**

**Admission requirements:** FMNN10 Numerical Methods for Differential Equations**Assumed prior knowledge:** FMAN35 Calculus in Several Variables and FMAN55 Applied Mathematics.

**Course coordinators:** **Web page:** https://www.ctr.maths.lu.se/course/NUMN28/