Syllabus

The following topics will be covered. Some will occupy more than one lecture. Lecture notes will be posted here after each class.

• Introduction and application of simulation to astrophysics
  A brief overview of the areas where simulation is used in astrophysics
• Broad overview of conservation laws (LeVeque Ch. 2)
  An introduction to the types of problems we will consider
• Derivation of the Euler equations (LeVeque Ch. 2, 14; Shu Ch. 2-4; Tritton Ch. 5)
  A physically motivated derivation of the equations of hydrodynamics
• Instabilities and Turbulence (Shu Ch. 8,9; Shore Ch. 9)
  A survey of the dominant instabilities in astrophysics
• A first look at shocks (Choudhuri Ch. 6)
  A high-level discussion of wave steepening and the derivation of the Rankine-Hugoniot jump conditions
• Numerical differentiation and Errors (Laney Ch. 6)
  A review of numerical differentiation, error norms, truncation error, and roundoff error
• ODEs
  A brief review of solving ordinary differential equations numerically
• Finite-volume methods
  more on the framework for solving hyperbolic PDEs
• Basics of Numerical Solutions (Ferziger & Peric Ch. 2)
  An overview of the properties that make up a good solution method
• Linear advection equation
  First order methods for solving a simple hyperbolic PDE.
• Explicit and implicit discretizations; stability analysis
  derivation of the CFL condition
• Higher-order methods
  Higher order reconstruction and slope limiters
• The Characteristic Form of the Euler Equations and Basics of Hyperbolic PDEs (Laney Ch. 3)
  
• Riemann problem
  The Riemann problem for the Euler equations and approximate solutions
• Solving the Euler equations
  Numerical methods for the equations of gas dynamics
• Case study: Type Ia supernovae
  A discussion of simulation techniques used in the study of SNe Ia
• Multidimensions
  Techniques for extending these methods to multiple dimensions
• Verification and validation
  How to know if you are getting the right solution.
• Software development practices
  Some tips and tools to help avoid common development mistakes
• Design of a hydro program
  An overview of the design of a modern hydrodynamics code
• High-performance computing
  Summary of the techniques used in high-performance computing, including MPI
• The FLASH code
  Tutorial on using the FLASH hydrodynamics code

Some optional topics that will be covered if there is time:

• Numerical Methods for the Heat Equation
  A brief discussion of some techniques used in solving the heat equation
• Multigrid
  The multigrid method for solving elliptic PDEs
• PPM
  An overview of the piecewise parabolic method
• Stellar Equation of State
  Extending methods to general equations of state
• Combustion
  Techniques used in combustion
• AMR
  An overview of adaptive mesh refinement
• Low speed flows
  Methods for low Mach number flows in astrophysics
• Cosmological hydrodynamics
  An introduction to the techniques used for cosmological simulation

Course work

There will be ~4 homework assigments throughout the term. Each student will give a ~15-20 minute presentation on the application of simulation to their field of interest. Additionally, these will be a final computational project, involving setting up, running, and analysing a fluid dynamics problem. Students will prepare a paper describing their results. The topic will be chosen together with the instructor.

updated Jan. 23, 2006