First part: statistics, basic finite-difference methods. Functions and interpolation, linear algebra, ordinary differential equations, partial differential equations.
Second part: spectral methods, conservative methods. Equations in conservative form and discontinuities. Riemann problem. Shock-capturing methods.
Laboratory numerical applications to problems of astrophysical interest.
W.H. Press et al. - Numerical recipes, Cambridge.
C.B. Laney – Computational gas dynamics, Cambridge.
Learning Objectives
Getting familiar with the elements of numerical analysis and computational fluid dynamics, in order to be able to write a numerical code of astrophysical interest and use existing codes.
Prerequisites
Basic informatics and programming. Fluid dynamics and dynamical processes in Astrophysics.
Teaching Methods
Frontal lectures (partly with the aid of electronic slides) and laboratory training.
Further information
Use of C/C++ and/or Fortran90 for coding and of Python libraries for output data visualisation.
Type of Assessment
Oral examination with discussion of a numerical code prepared by the candidate.
Course program
Basic finite-difference methods. Functions and interpolation: Sturm-Liouville analysis, Fourier analysis, splines. Linear algebra: solution of linear systems, iterative methods. Ordinary differential equations: stability, consistency, convergence, one or multiple steps methods, higher order methods. Partial differential equations: methods for elliptic (Laplace’s and Poisson’s problems) and parabolic equations; methods for hyperbolic equations. Spectral methods.
Second part: conservative methods. Shock formation and shock-capturing numerical methods for nonlinear hyperbolic equations. Finite differences, central and upwind methods. Riemann problem, characteristic waves; Godunov , Roe and Lax-Friedrichs methods. Applications to gas dynamics and magneto hydrodynamics. Simple benchmark problems and specific applications of astrophysical interest.