# Numerical Methods in Astrophysics

Computation has become an essential tool in astrophysics research (e.g. numerical analysis, modelling, data analysis, detector calibration). Researchers at Astrophysics Research institute of Liverpool John Moores University use scientific computation to study an enormous range of physical processes. At large scales, N-body and hydrodynamics methods are used to study cosmological structure formation and galaxy formation. At smaller scales, a wide variety of numerical methods are used to understand stellar dynamics, stellar evolution, core-collapse supernovae, gamma-ray bursts and relativistic jets. In this module, three world-leading theoretical astrophysicists discuss numerical methods with an emphasis on their applications to real astrophysics research.

The first two lectures deal with random number problems, such as techniques to generate random numbers with a desired probability distributions, Monte Carlo integration, and its applications to astronomy studies. Students will work on project (1) Monte Carlo simulations to determine biases in absolute magnitudes of stars retrieved from their parallaxes. The next two lectures discuss the integration of ordinary differential equations and Fast Fourier transform method. Students will study the following two stellar dynamics projects and use the techniques which are discussed in the lectures: project (2) Calculation of closed periodic orbits and orbits oscillating around them, and project (3) Determination of regularity of orbits found in Project 2 using Fast Fourier Transform. The last set of lectures expand the material discussed in the previous lectures, and discuss the applications to astrophysical problems. These lectures will be associated with two projects: project (4) random walk and diffusion process, and project (5) three-body problem and hypervelocity stars.

Topics covered in the course

- Random numbers

- Techniques to generate random numbers with uniform probability distribution (pdf)

- Methods to generate random numbers with other pdfs

- MC integration

- Methods for deterministic numerical integration

- Comparison with deterministic integration

- Methods to improve convergence of MC integration

- Integration of ordinary differential equations

- Runge-Kutta method

- Integration of equations of motion: 2nd order differential equations

- Example trajectories in a fixed gravitational potential

- Fast Fourier Transform

- Application to discrete data sets

- Example Fourier Transforms of regular and chaotic orbits

- Random walk and diffusion process

- Partial differential equations: diffusion equations

- Units and normalisations in numerical calculations

- Finite difference methods

- von Neumann stability analysis

- Supermassive black holes and tidal disruption events

- Hill's mechanism, S-stars around the Galactic centre and hypervelocity stars