48716 - Numerical Calculation and Statistics

Learning outcomes

At the end of the class the student will be able to numerically solve ordinary and partial derivative equations (for instance the gasdynamics equations). The student will be able to set-up and investigate relevant hydrodynamical problems, like the evolution of SN remnants, AGN feedback in galaxies, and more).

Moreover, the student will get familiar with modern N-body codes and will solve simple stellar dynamical problems.

Course contents

 

Part 1 (Prof. Fabrizio Brighenti)

 Gasdynamics: 1) astrophysical plasma, 2) equations of gasdynamics; 3) sound waves and shock waves; 4) astrophysical applications: evolutions of supernova remnants, stellar wind bubbles, galactic winds, Bondi accretion, AGN feedback.

Numerical Astrophysics. 1) finite differences; consistence, convergence and stability of a numerical scheme. 2) Conservative and transportative properties, control volume approach.. 3) Methods for hyperbolic equations: FTCS, Lax, Upwind. Von Neuman stability analysis. 4) Second order methods for hyperbolic equations: Lax-Wendroff, Upwind II order. 5) Methods for parabolic equations: explicit and implicit schemes. Crank-Nicholson method. 6) Methods for Hydrodynamics: a simple 1d hydrocode. 7) Tests and applications of the hydrocode: Sod shock tube, SNRs, stellar wind bubbles, cooling flows, Bondi accretion...

Lab Projects: 1) Shock tube test; 2) Evolution of SN remnants and stellar wind bubbles; 3) AGN feedback

 

Part 2 (Prof. Federico Marinacci)

1) Introduction to the structure of hydrocodes. Eulerian and Lagrangean codes. Conservative discretization of Euler equations.

2) Godunov methods. Numerical flux concept and fluxes as solutions of the local Riemann problems. Solution of the Riemann problem for Euler equations. Riemann solvers and approximate solutions for numerical fluxes.

3) Higher order Godunov methods. II order TVD schemes. Time-step calculation. Extrapolation of variable at the cell interfaces to II order accuracy. Radiative cooling.

4) Use of (II order) Godunov methods to calculate the SNR evolution: adiabatic and radiative cases.

Readings/Bibliography

Lecture slides (on Campus website)

 

Text on the fluid dynamics part:

Clarke & Carswell: “Principles of Astrophysical Fluid Dynamics”

 

Texts on the numerical fluid dynamics part:

Numerical Methods in Astrophysics”, Bodenheimer et al.

Numerical Recipes”, Press et al.

Teaching methods

Lectures on the theory used, using both slides and blackboard. Lab sessions.

Assessment methods

Oral exam and written report on the astrophysical project investigated during the Lab sessions.

Office hours

See the website of Fabrizio Brighenti

See the website of Federico Marinacci