Summersemester, 2009

C.P. Dullemond and H.H. Wang

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.

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Topics:

In this lecture we will discuss modern numerical algorithms for solving the equations of gas- or fluid dynamics in the terrestrial as well as astrophysical context. We first discuss the fundamental equations to be solved: the Euler equations and its mathematical properties. Then we discuss numerical methods for advection on a 1-D grid. We discuss how such advection algorithms are used for solving the equations of hydrodynamics in 1-D and 2-D using classical methods. Then we proceed 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 discuss methods of implicit integration. These methods allow the solution of viscous hydrodynamics problems with extremely large dynamic range. We briefly discuss methods used for solving incompressible fluid dynamics, such as the flow of water in a pipe system or the flow of ocean currents. We then make a brief excursion to methods of Finite Elements for modeling solids. Returning to astrophysics we end with discussing an alternative method for solving hydrodynamics equations: Smooth Particle Hydrodynamics (SPH).

On-line chapters:

• Introduction
• Chapter 1: The equations of fluid dynamics
• Chapter 2: Hyperbolic equations
• Chapter 3: Numerical advection (I)
• Chapter 4: Numerical advection (II)
• Chapter 5: Classical hydrodynamic algorithms
• Chapter 6: Riemann solvers I
• Chapter 7: Riemann solvers II
• Chapter 8: Implicit differencing and incompressible fluids
• Chapter 9: Modeling solids
• Chapter 10: Non-cartesian coordinate systems
• Chapter 11: Smooth particle hydrodynamics
• Appendix B: IDL / GDL Micro-manual and exercises

Here are the exercise sheets:

• Problem sheet 1
• Problem sheet 2
• Problem sheet 3
• Problem sheet 4
• Problem sheet 5
• Problem sheet 6
• Problem sheets 7 and 8
• Problem sheet 9

Here is an IDL example of a simple FEM model: fem2d.pro, fem2d_bridge.pro.

A note on newer versions of this lecture

This 2009 lecture was the last version that I gave by alone (with of course my student assistants). In 2010 this lecture was not given. As of 2011 this lecture is given together with Volker Springel. That lecture has a new focus, in particular in the second half. See webpage for the 2011 version of this lecture.

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"
• Felippa, Carlos A., "Introduction to Finite Element Methods", http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/