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/*@@
@file GRHydro_Bondi.F90
@date Wed Jan 13 13:00:49 EST 2010
@author Scott C. Noble
@desc
Hydro initial data for the relativistic Bondi solution about
a single Schwarzschild black hole.
@enddesc
@@*/
/*
Calculates the Bondi solution, or the spherically symmetric hydrostationary
solution to a fluid on a static fixed background spacetime. We assume that one can
calculate a radius "r" from the grid and that with respect to this radial coordinate,
the solution satisfies
d (\rho u^r) / dr = 0
Assumes that the equation of state is P = K \rho^\Gamma and K is set by
the location of the sonic point.
-- Implicitly assumes that there is no spin in the geometry as there is no Bondi
solution for spinning black holes. If a spin is specified, a spherically symmetric
is still assumed but the 4-velocity is set consistently with the spinning spacetime.
*/
#include "cctk.h"
#include "cctk_Arguments.h"
#include "cctk_Parameters.h"
#include "cctk_Functions.h"
#include "GRHydro_Macros.h"
# define M_PI 3.14159265358979323846d0 /* pi */
!!$Newton-Raphson parameters:
#define velx(i,j,k) vel(i,j,k,1)
#define vely(i,j,k) vel(i,j,k,2)
#define velz(i,j,k) vel(i,j,k,3)
#define sx(i,j,k) scon(i,j,k,1)
#define sy(i,j,k) scon(i,j,k,2)
#define sz(i,j,k) scon(i,j,k,3)
subroutine GRHydro_Bondi_Iso(CCTK_ARGUMENTS)
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_PARAMETERS
DECLARE_CCTK_FUNCTIONS
CCTK_INT :: i, j, k, nx, ny, nz, imin, jb,N_points
CCTK_REAL :: ONEmTINY
PARAMETER (N_points=2000,ONEmTINY=0.999999d0)
CCTK_REAL :: M, Msq, Mdot, rs, gam, rmin_bondi, rmax_bondi, cs_sq,cs,vs_sq,vs,rhos,gtemp,hs, Kval, Qdot
CCTK_REAL :: logrmin,dlogr,rhotmp,utmp,vtmp,rspher
CCTK_REAL :: r_bondi(N_points), logr_bondi(N_points), rho_bondi(N_points), u_bondi(N_points), v_bondi(N_points)
CCTK_REAL :: drhodr, det, rhocheck, rhocheck2, riso, rnew, rsch, ucheck, psinew
CCTK_REAL :: uiso, uisocheck, vcheck, ucheck2, vcheck2, xhat,yhat, zhat, xp, yp, zp
CCTK_REAL :: f,df,ddf,a,b,c,rsm,roverm
!!$set_bondi_parameters
M = bondi_central_mass(1)
Msq = M*M
Mdot = mdot_sonicpt_bondi
rs = r_sonicpt_bondi
gam = gl_gamma
write(*,*) 'Bondi_pars:',M,mdot_sonicpt_bondi,r_sonicpt_bondi,gl_gamma
rmin_bondi = M * bondi_rmin(1)
rmax_bondi = M * bondi_rmax(1)
cs_sq = M / ( 2.*rs - 3.*M )
if( cs_sq > (gam - 1.)) then
cs_sq = gam - 1.
rs = 0.5 * M * ( 3. + 1./cs_sq )
endif
cs = sqrt(cs_sq)
vs_sq = M / ( 2. * rs )
vs = sqrt(vs_sq)
rhos = Mdot / ( 4. * M_PI * vs * rs * rs )
gtemp = gam - 1.
hs = 1. / ( 1. - cs_sq / (gam - 1.) )
Kval = hs * cs_sq * rhos**(-gtemp) / gam
Qdot = hs * hs * ( 1. - 3. * vs_sq )
logrmin = log10(rmin_bondi)
dlogr = (log10(rmax_bondi) - logrmin)/(1.*(N_points-1))
write(*,*)'More pars:',cs,vs,rhos,hs,Kval,Qdot,logrmin,dlogr
rhotmp=1.0d30
imin=1
do i=1,N_points
logr_bondi(i) = logrmin + dlogr*(i-1)
r_bondi(i) = 10.**(logr_bondi(i))
utmp = abs(r_bondi(i) - r_sonicpt_bondi)
if (utmp < rhotmp) then
rhotmp = utmp
imin = i
endif
enddo
rhotmp = -1. !!$ start with guess
do i=imin,N_points
rspher = r_bondi(i)
call find_bondi_solution( rspher, rhotmp, utmp, vtmp, rs, rhos, M, Mdot, Kval, gam, Qdot )
if(rhotmp < initial_rho_abs_min) then
rhotmp = initial_rho_abs_min
utmp = Kval * rhotmp**gl_gamma / (gl_gamma - 1.)
endif
rho_bondi(i) = rhotmp
u_bondi(i) = utmp
v_bondi(i) = vtmp
end do
rhotmp = -1.
do i=imin-1,1,-1
rspher = r_bondi(i)
call find_bondi_solution( rspher, rhotmp, utmp, vtmp, rs, rhos, M, Mdot, Kval, gam, Qdot )
if(rhotmp < initial_rho_abs_min) then
rhotmp = initial_rho_abs_min
utmp = K * rhotmp**gl_gamma / (gl_gamma - 1.)
endif
rho_bondi(i) = rhotmp
u_bondi(i) = utmp
v_bondi(i) = vtmp
enddo
write(*,*)"i=1:",r_bondi(1),rho_bondi(1),u_bondi(1),v_bondi(1)
write(*,*)"i=100:",r_bondi(100),rho_bondi(100),u_bondi(100),v_bondi(100)
write(*,*)"i=1000:",r_bondi(1000),rho_bondi(1000),u_bondi(1000),v_bondi(1000)
write(*,*)"i=1500:",r_bondi(1500),rho_bondi(1500),u_bondi(1500),v_bondi(1500)
!!$ // find the derivative near r=M
rnew = 2.25 * M
j = floor ( 0.5 + (log10(rnew) - logrmin) / dlogr )
rhocheck = rho_bondi(j)
call find_bondi_solution(rnew,rhocheck, ucheck, vcheck, rs, rhos, M, Mdot, Kval, gam, Qdot )
uisocheck = 4.0*vcheck/3.0
rnew = 0.25 * 3.02**2 * M/1.01
j = floor( 0.5 + (log10(rnew) - logrmin) / dlogr )
rhocheck2 = rho_bondi(j)
call find_bondi_solution( rnew, rhocheck2, ucheck2, vcheck2, rs, rhos, M, Mdot, Kval, gam, Qdot )
drhodr = 100.0*(rhocheck2-rhocheck)/M
write(6,*)'Rhocheck:',rhocheck,rhocheck2,drhodr
nx = cctk_lsh(1)
ny = cctk_lsh(2)
nz = cctk_lsh(3)
do i=1,nx
do j=1,ny
do k=1,nz
xp=x(i,j,k)
yp=y(i,j,k)
zp=z(i,j,k)
riso = sqrt(xp*xp + yp*yp + zp*zp +1.0e-16)
xhat = xp/riso
yhat = yp/riso
zhat = zp/riso
roverm = riso/M
if(roverm > ONEmTINY) then
rsch = 0.25 * ( 2.*riso + M)**2 / riso
jb = floor( 0.5 + (log10(rsch) - logrmin) / dlogr )
if(jb > N_points)jb = N_points
rhotmp = rho_bondi(jb)
call find_bondi_solution( rsch,rhotmp, utmp, vtmp, rs, rhos, M, Mdot, Kval, gam, Qdot)
rho(i,j,k) = rhotmp
uiso = vtmp / (1.0 - M/2.0/riso) / (1.0+ M/2.0/riso)
else
if(roverm > 0.5d0*ONEmTINY) then
rho(i,j,k) = rhocheck+drhodr*riso*(riso-M)/M
else
rho(i,j,k) = (rhocheck-drhodr*M/4.0)*(1.-cos(2.*M_PI*riso/M))/2.0
endif
utmp = Kval * rho(i,j,k)**( gam ) / (gam - 1.)
uiso = uisocheck * riso / M
endif
eps(i,j,k) = utmp/rhotmp
w_lorentz(i,j,k) = sqrt(1.0+gxx(i,j,k) * uiso**2)
velx(i,j,k) = -1.0*uiso * xhat / w_lorentz(i,j,k)
vely(i,j,k) = -1.0*uiso * yhat / w_lorentz(i,j,k)
velz(i,j,k) = -1.0*uiso * zhat / w_lorentz(i,j,k)
det=SPATIAL_DETERMINANT(gxx(i,j,k),gxy(i,j,k),gxz(i,j,k),gyy(i,j,k),gyz(i,j,k),gzz(i,j,k))
call Prim2ConGen(GRHydro_eos_handle,gxx(i,j,k),gxy(i,j,k), &
gxz(i,j,k),gyy(i,j,k),gyz(i,j,k),gzz(i,j,k), &
det, dens(i,j,k),sx(i,j,k),sy(i,j,k),sz(i,j,k), &
tau(i,j,k),rho(i,j,k), &
velx(i,j,k),vely(i,j,k),velz(i,j,k), &
eps(i,j,k),press(i,j,k),w_lorentz(i,j,k))
if(riso.gt.1.014d0.and.riso.lt.1.015)write(6,*)'Point to check:', &
x(i,j,k),y(i,j,k),z(i,j,k),riso,gxx(i,j,k),dens(i,j,k),tau(i,j,k),&
sx(i,j,k),sy(i,j,k),sz(i,j,k),rho(i,j,k),eps(i,j,k),&
velx(i,j,k),vely(i,j,k),velz(i,j,k)
end do
end do
end do
densrhs = 0.d0
srhs = 0.d0
taurhs = 0.d0
return
end subroutine GRHydro_Bondi_Iso
subroutine find_bondi_solution(r, rho, u, v, rs, rhos, M, Mdot, Kval, gam, Qdot )
implicit none
CCTK_REAL :: r, rho, u, v, rs, rhos, M, Mdot, Kval, gam, Qdot
CCTK_REAL :: ur,r_sol, rho_old
CCTK_REAL :: f, df, dx, x_old, resid, jac
CCTK_REAL :: errx, x_orig, x
CCTK_INT :: n_iter, i_extra, doing_extra, keep_iterating, i_increase
CCTK_REAL :: vp, h, hp, term
CCTK_REAL :: newt_tol_b,small_bondi
CCTK_INT :: max_newt_iter_b, extra_newt_iter_b
max_newt_iter_b = 30
newt_tol_b = 1.0e-15
extra_newt_iter_b = 2
small_bondi = 1.0e-20
!!$ if(r>8.1043 .and. r<8.1044)write(*,*)'init guess:',r,rho
!!$ write(*,*)'init guess:',r,rho
if (rho < 0.) then
if( r > 0.9*rs .and. r < 1.1*rs ) then
rho = rhos
else
if(r < rs) then
ur = r**(-0.5)
else
ur = 0.5*r**(-1.5)
endif
rho = Mdot / (4.*M_PI * r * r * ur)
endif
endif
!!$ if(r>8.1043 .and. r<8.1044)
!!$ write(*,*)'init guess:',r,ur,rho,rs,rhos,Mdot
!!$ if(r<1.0001e-10)write(*,*)'init guess:',r,ur,rho,rs,rhos,Mdot
!!$ write(*,*)'init guess:',r,ur,rho,rs,rhos,Mdot
!!$ set global variables needed by residual function:
r_sol = r
!!$ Use Newton's method to find rho:
!!$ gnr_bondi( rho, NEWT_DIM_B, bondi_resid)
errx = 1.0
df=1.0
f=1.0
doing_extra = 0
rho_old=rho
x=rho
n_iter = 0
!!$ Start the Newton-Raphson iterations :
keep_iterating = 1
do while( keep_iterating == 1 )
hp = Kval * gam * x**(gam - 2.) !!$ // dh/drho
h = 1. + hp * x / ( gam - 1. )
v = Mdot / ( 4. * M_PI * r_sol * r_sol * x )
vp = -v / x !!$ // dv/drho
term = 1. - 2.*M/r_sol + v*v
resid = -Qdot + h * h * term
jac = 2. * h *( hp*term + h*v*vp )
dx = -resid / jac
f = 0.5*resid**2
df = -2. * f
/* Save old values before calculating the new: */
errx = 0.
x_old = x
x=x+dx
!!$ if(r>8.1043 .and. r<8.1044)write(*,*)'iter:',x,dx,resid,jac
!!$ /****************************************/
!!$ /* Calculate the convergence criterion */
!!$ /****************************************/
!!$ /* For the new criterion, always look at relative error in indep. variable: */
!!$ // METHOD specific:
if(x==0) then
errx = abs(dx)
x = small_bondi
else
errx = abs(dx/x)
endif
!!$ /*****************************************************************************/
!!$ /* If we've reached the tolerance level, then just do a few extra iterations */
!!$ /* before stopping */
!!$ /*****************************************************************************/
!!$ if(r>8.1043 .and. r<8.1044)write(*,*)'iter3:',errx,newt_tol_b,keep_iterating, &
!!$ doing_extra,i_extra
if((abs(errx)<=newt_tol_b) .and. (doing_extra == 0) .and. (extra_newt_iter_b > 0)) &
doing_extra=1
if( doing_extra == 1 ) i_extra=i_extra+1
if( ((abs(errx) <= newt_tol_b).and.(doing_extra == 0)) .or. &
(i_extra > extra_newt_iter_b) .or. (n_iter >= (max_newt_iter_b-1)) ) &
keep_iterating = 0
!!$ if(r>8.1043 .and. r<8.1044)write(*,*)'iter4:',errx,newt_tol_b,keep_iterating, &
!!$ doing_extra,i_extra
n_iter=n_iter+1
end do
rho=x
!!$ Calculate other quantities:
u = Kval * rho**( gam ) / (gam - 1.)
v = Mdot / ( 4. * M_PI * r * r * rho )
!!$ if(r>8.1043 .and. r<8.1044)
!!$ write(*,*)'final:',r,rho,u,v
return
end subroutine find_bondi_solution
|
