Advanced Computational Astrophysics: Concepts and Applications

Nearly all theoretical astrophysics research as well as a substantial part of observational astronomy requires an expert knowledge of computational techniques. The topics range from ordinary and partial differential equations which find applications in the laws of gravitation, hydrodynamics or diffusion, to Fourier techniques, to collisionless systems. The objectives of this course are to endow students with the capacity to identify and classify common numerical problems in modern astrophysics. The course aims at an active understanding of numerical methods and algorithms, their ranges of applicability and teaches the underlying physical concepts. Solving basic astrophysical problems with adequate numerical techniques and determining the range of validity is an essential part of the course.

Lecturers:

Place and Dates:

PDF file

Contents:

• Reviewing basic concepts of numerical simulations (preparatory course)
  
  • Integer arithmetic
    
  • Floating point arithmetic
    
    
• Integration of ordinary differential equations
  
  • The Euler method: explicit and implicit
    
  • Runge-Kutta methods and adaptive step sizes
    
  • The leapfrog and symplectic integrators
    
    
• Basic gas dynamics
  
  • Basic conservation laws
    
  • Euler and Navier-Stokes equations
    
  • Shocks, fluid instabilities and turbulence
    
    
• Eulerian hydrodynamics
  
  • Types and solution schemes of partial differential equations
    
  • Advection and the Riemann problem
    
  • Finite volume discretization
    
  • Godunov method and Riemann solvers
    
  • Extensions to multiple dimensions and high-order accuracy
    
    
• Smoothed particle hydrodynamics (SPH)
  
  • Kernel interpolants
    
  • Variational derivation of SPH
    
  • Artificial viscosity
    
  • Advantages and disadvantages of SPH
    
    
• Diffusion equation
  
  • Explicit method
    
  • Implicit methods
    
    
• Fourier transform techniques
  
  • Convolution problems
    
  • The discrete Fourier transform
    
  • Non-periodic problems with 'zero padding'
    
    
• Collisionless particle systems
  
  • N-particle ensembles
    
  • Uncorrelated (collisionless) systems
    
  • N-body models of collisionless systems
    
    
• Tree algorithms
  
  • Multipole expansion
    
  • Hierarchical grouping
    
  • Tree walk
    
    
• The particle-mesh technique
  
  • Mass/charge assignment
    
  • Solving for the potential and force calculation
    
  • Interpolating from the mesh to the particles
    
    
• Paralellization techniques
  
  • Hardware overview
    
  • Shared memory parallelization with OpenMP
    
  • Distributed memory parallelization with MPI
    
    

Homework Assignments

Python Preparatory Course:

Credit Points: