# Numerical Fluid Mechanics

### Summersemester, 2008

### C.P. Dullemond and R. Kuiper

Movie: hydrodynamic simulation of a supersonic jet-stream injected into a homogeneous medium. The computation is done on a 100x100 grid using Roe's algorithm. |

## Topics:

In this lecture we discussed modern numerical algorithms for solving the equations of gas- or fluid dynamics in the terrestrial as well as astrophysical context. We first discussed the fundamental equations to be solved: the Euler equations and its mathematical properties. Then we discussed numerical methods for advection on a 1-D grid. We discussed how such advection algorithms are used for solving the equations of hydrodynamics in 1-D and 2-D using classical methods. Then we proceeded with modern Riemann solver algorithms, including Roe's method, HLLE, HLLC, PPM solvers etc., i.e. the kind of solvers used in many of the newest astrophysical numerical hydrodynamics packages. We discussed methods of*implicit integration*. These methods allow the solution of viscous hydrodynamics problems with extremely large dynamic range. We briefly discussed methods used for solving incompressible fluid dynamics, such as the flow of water in a pipe system or the flow of ocean currents. Returning to astrophysics we discussed alternative methods for solving hydrodynamics equations such as Smooth Particle Hydrodynamics.

## About the lecture:

This lecture has ended on Monday Jul 15. The material remains ready for download for those who are interested. No warranties. The material is "as is", and I do not accept any responsibility for what readers may do with the information in these chapters. The lecture is meant for students who have an interest in learning how real gas/fluid dynamics problems are solved in astrophysics and enigineering sciences. Knowledge of linear algebra, vector calculus and classical mechanics are required, as well as basic programming skills.

## On-line chapters:

For the 2008 version of this course the on-line chapters are no longer available. Please have a look at the 2009 version of this lecture.## On-line exercises:

The exercises are managed by Rolf Kuiper, who has set up a web site for the exercises (no longer available).

## Literature:

The above lecture notes are for significant parts inspired by the following books:- Randall J. LeVeque, "Finite Volume Methods for Hyperbolic Problems"
- Bodenheimer, Laughlin, Rozyczka and Yorke, "Numerical Methods in Astrophysics: An Introduction"
- Toro, E.F. "Riemann Solvers and Numerical Methods for Fluid Dynamics" (Springer Verlag)
- Collela and Puckett, "Modern Numerical Methods for Fluid Flow"
- H. W. Liepmann and A. Roshko, "Elements of Gas Dynamics"
- Ferziger, J. H. and Peric, M, "Computational Methods for Fluid Dynamics"

Responsible: Cornelis Petrus Dullemond, last modification Jul/28/2012 23:11 CEST