Locally inertial approximations of balance laws arising in (1+1)-dimensional general relativity

Laurent Gosse (1)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"

An elementary model of $1+1$-dimensional general relativity, known as ``$\R=\T$'' and mainly developed by Mann {\it et al.} \cite{Mbh1,Mbh2,MMSS,MST,MR,MM,SM}, is set up in various contexts. Its formulation, mostly in isothermal coordinates, is derived and a relativistic Euler system of self-gravitating gas coupled to a Liouville equation for the metric's conformal factor is deduced. First, external field approximations are carried out: both a Klein-Gordon equation is studied along with its corresponding density, and a Dirac one inside an hydrostatic gravitational field induced by a static, piecewise constant mass repartition. Finally, the coupled Euler-Liouville system is simulated, by means of a locally inertial Godunov scheme: the gravitational collapse of a static \underline{random} initial distribution of density is displayed. Well-balanced discretizations rely on the treatment of source terms at each interface of the computational grid, hence the metric remains flat in every computational cell.

Mots clés: Locally inertial scheme: Relativistic hydrodynamics; Structure-Preserving and Well-balanced schemes.; Intrinsic finite-differences; 1+1 general relativity; Dirac and Klein-Gordon equations

Domaines: Analyse numérique [math.NA]

Laurent Gosse. Locally inertial approximations of balance laws arising in (1+1)-dimensional general relativity. SIAM Journal on Applied Mathematics, 2015, ⟨10.1137/140969889⟩. ⟨hal-01119168⟩