Outflow

Roland Haas <roland.haas@physcis.gatech.edu>

August 15 2009

Abstract

Outflow calculates the flow of rest mass density across a SphericalSurface, eg. and apparent horizon or a sphere at “infinity”.

1 Introduction

Hydrodynamic simulations should conserve eg. the total rest mass in the system (outside he event horizon, that is). This thorn allow to measure the flux of rest mass across a given SphericalSurface.

2 Physical System

The Valencia formalism defines its D = ρW variable such that one can define a total rest mass density as:

M =γDd3x

and from the EOM of D (see [1])

x0(γD) + xi((γ)αD(vi βiα) = 0

one obtains:

= V xi(γαD(vi βiα)d3x (1) = V γαD(vi βiα)dσ i (2)

where σi is the ordinary flat space directed surface element of the enclosing surface, eg. 

dσi = r̂ir2 sin𝜃d𝜃dϕ

with r̂i = [cosϕsin𝜃,sinϕsin𝜃,cos(𝜃)] for a sphere of radius r.

For a generic SphericalSurface parametrized by 𝜃 and ϕ one has:

x = r̄(𝜃,ϕ)cosϕsin𝜃 (3) y = r̄(𝜃,ϕ)sinϕsin𝜃 (4) z = r̄(𝜃,ϕ)cos𝜃 (5)

where r̄ is the isotropic radius. Consequently the surface element is

dσi = r̄ 𝜃 ×r̄ ϕ i (6) = r̄2 sin𝜃r̄̂ i r̄ 𝜃 r̄sin𝜃𝜃̂i r̄ ϕ r̄ϕ̂i (7)

where r̄̂, 𝜃̂ and ϕ̂ are the flat space standard unit vectors on the sphere [?].

3 Numerical Implementation

We implement the surface integral by interpolating the required quantities (gij, ρ, vi, βi, α) onto the spherical surface and then integrate using a fourth order convergent Newton-Cotes formula.

For the 𝜃 direction SphericalSurfaces defines grid points such that

𝜃i = (n𝜃 12)Δ𝜃 + iΔ𝜃0 i < N𝜃 1

where N𝜃 is the total number of intervals (sf_ntheta), n𝜃 is the number of ghost zones in the 𝜃 direction (nghosttheta) and Δ𝜃 = π N𝜃2n𝜃. Since with this

𝜃i [Δ𝜃2,π Δ𝜃2]for i [n𝜃,N𝜃 n𝜃 1]

we do not have grid points at the end of the interval {0,π} we derive an open extended Newton-Cotes formula from Eq. 4.1.14 of [2] and a third order accurate extrapolative rule (see Maple worksheet). We find

x0xN1 f(x)dx h{13 12f12 + 7 8f32 + 25 24f52 + f72 + f92 + + fN172 +fN192 + 25 24fN152 + 7 8fN132 + 13 12fN112} + O(1N4) (8)

.

For the ϕ direction SphericalSurfaces defines grid points such that

ϕi = nϕΔϕ + iΔϕ0 i < Nϕ 1

where Nϕ is the total number of intervals (sf_nphi), nϕ is the number of ghost zones in the ϕ direction (nghostphi) and Δϕ = π Nϕ2nϕ. With this

ϕi [0,2π Δϕ]for i [nϕ,Nϕ nϕ 1]

we use a simple extended trapezoid rule to achieve spectral convergence due to the periodic nature of ϕ (note: xN = x0)

x0xN f(x)dx h i=0N1fi (9)

.

The derivatives of r̄ along 𝜃 and ϕ are obtained numerically and require at least two ghost zones in 𝜃 and ϕ.

4 Using This Thorn

Right now surface can only be prescribed by SphericalSurfaces, the flux through each surface is output in in a file outflow_det_%d.asc

4.1 Interaction With Other Thorns

Takes care to schedule itself after SphericalSurfaces_HasBeenSet.

4.2 Examples

See the parameter file in the test directory. For spherical symmetric infall the flux through all detectors should be equal (since rest mass must be conserved).

References

[1]   José A. Font, “Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity”, Living Rev. Relativity 11, (2008), 7. URL (cited on August 15. 2009): http://www.livingreviews.org/lrr-2008-7