First part: basic finite-difference methods. Functions and interpolation, linear algebra, ordinary differential equations, partial differential equations.
Second part: conservative methods. Equations in conservative form and discontinuities. Riemann problem. Shock-capturing methods.
Laboratory numerical applications to problems of astrophysical interest.
To acquire the bases of numerical analysis and of computational gasdynamics, and to be able to write a numerical code of astrophysical interest.
Prerequisites
Basic informatics and programming. Fluid dynamics and dynamical processes in Astrophysics.
Teaching Methods
6 CFU, 48 hours. Two modules of 3 CFU. Frontal lectures (partly with the aid of electronic slides) and laboratory training.
Further information
Pdf lectures provided to students.
Type of Assessment
Oral examination with discussion of a numerical code prepared by the candidate.
Course program
First part: basic finite-difference methods. Functions and their 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.
Second part: conservative methods. Introduction to 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 gasdynamics, magneto-hydrodynamics (MHD), and relativistic hydrodynamics.
Numerical codes: implementation and basic parallel methods on multi-processor platforms. Laboratoty numerical applications to problems of astrophysical interest.