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Code construction – 4: Surface fitted coordinate for the fluid and hydrostatic eq.

Code construction – 4: Surface fitted coordinate for the fluid and hydrostatic eq. On top of the spherical coordinates for the field (r, q , f ), we introduce surface fitted coordinates for the fluid as follows. where R( q , f ) is the surface of a star.

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Code construction – 4: Surface fitted coordinate for the fluid and hydrostatic eq.

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  1. Code construction – 4: Surface fitted coordinate for the fluid and hydrostatic eq. On top of the spherical coordinates for the field (r, q, f), we introduce surface fitted coordinates for the fluid as follows. where R(q,f) is the surface of a star. The surface R(q, f) is normalized by the value at q = p/2, and f = 0. R0 := R(p/2,0). subroutine interpolation_grav_to_fluid q, and f coordinates are the same for the field and fluid coordinates. So, an interpolation along the radial coordinate (1D interpolation) is required. It is recommended to use an interpolating formula higher than the 2nd order. rgi-1rgi r0 rfj-1 rfj end interpolation_grav_to_fluid

  2. subroutine coordinate_patch_kit_fluid We use the same grid spacing for the fluid coordinate (rf , qf ,ff) as the coordinate grids for the gravitational field. Radial coordinate input parameters: nrf. (rf = rg up to rf = 1) R(q,f) : rf2 [0, 1], the grid spacing rule is the same as rg. nrf : Total number of radial grid points (-1). rf(ir), ir = 0, nrf : Radial coordinate grid points ri . hrf(ir) : Mid-point of radial grids. drf(ir) : Radial grid spacing Dri := ri – ri-1. q coordinate input parameter: ntf. q2 [0, p], the grid spacing is equidistant. thf(it), it = 0, ntf : q grid points qi . hthf(it) : Mid-point of q gridsqi+1/2. dthf : q grid spacing Dq. ntf : Total number of q grid points (-1). Functions associated with q coordinate. Trigonometric funcitons. sinthf(it), it = 0, ntf : sinqi. hsinthf(it) : sinqi+1/2 sin at mid-point of q grids. costhf(it), it = 0, ntf : cosqi. hcosthf(it) : cosqi+1/2 cosin at mid-point of q grids.

  3. f coordinate input parameter: npf. phif(ip), ip = 0, npf : f grid points fi . hphif(ip) : Mid-point of f gridsfi+1/2. dphif : f grid spacing Df. f2 [0, 2p], the grid spacing is equidistant. npf : Total number of f grid points (-1). Functions associated with f coordinate. Trigonometric funcitons. sinphif(ip), ip = 0, npf : sinfi. hsinphif(ip) : sinfi+1/2 sin at mid-point of f grids. cosphif(ip), ip = 0, npf : cosfi. hcosphif(ip): cosfi+1/2 cos at mid-point of f grids. Weight for the integration: assigned at the mid-points. wrf(irr) = hrf(irr)2 drf(irr) : weight for the radial integratoin ri2Dri wtf(itt) = hsinthef(itt) dthf : weight for the q integratoin sinqi Dqi wpf(ipp) = dphif : weight for the f integratoin Dfi wrtpf(irr,itt,ipp) = hR(itt,ipp)3 £ wrf(irr) £ wtf(itt) £ wpf(ipp) hR(itt,ipp) is the radius of the stellar surface at the mid-points of (q, f) grids. That is, at hth(itt), and hphi(ipp) . end subroutine coordinate_patch_kit_fluid

  4. subroutine rotating_compact_star Interpolate which are used to compute ut. call interpolation_grav_to_fluid call hydrostatic_equation call update_matter call update_surface call update_parameter Same as the gravitational field (a different value for l might make a convergence faster.) end subroutine rotating_compact_star

  5. Recall: polytropic (adiabatic) EOS.

  6. subroutine update_parameter Same as the Newtonian calculation, we have three parameters, Three conditions are imposed at the center, and two points at the surface. When the length scale is updated from Ri to R0 , the lapse and the conformal factor changes as, (When an iteration is made, the physical size of a star may change. While in numerical computation, we normalize the size (to set R(p/2,0)=1) and update the length scale R0.) Three conditions h=hc at the center, h=1 at Req and Rp, are applied to and solved for the parameters . end subroutine update_parameter

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