aproximación CFC

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En la discretización que hicimos teníamos dos sistemas acoplados, uno para las $latex X^i$ y otro para las $latex beta^i$. Procedemos ahora a desacoplarlos.

Para empezar, tomamos la divergencia (plana) del sistema:

$latex Delta X^i = 8 pi f^{ij} S^*_j – frac{1}{3}mathcal{D}^i mathcal{D}_j X^j$

y, teniendo en cuenta que $latex mathcal{D}$ conmuta con $latex Delta$ (métrica plana), tenemos:

$latex Delta (mathcal{D}_i X^i) = 8 pi mathcal{D}^j S^*_j – frac{1}{3} Delta (mathcal{D}_j X^j)$,

por lo que:

$latex Delta (mathcal{D}_i X^i) = frac{3}{4} 8 pi mathcal{D}^j S^*_j$.

De esta manera, si definimos $latex Theta_X := mathcal{D}_i X^i$, nos queda:

$latex Delta Theta_X = frac{3}{4} 8 pi mathcal{D}^j S^*_j = 6 pi (partial_x S^*_x + partial_y S^*_y +partial_z S^*_z )$,

que discretizado queda:

$latex frac{(Theta_X)_{i-1,j,k}-2(Theta_X)_{i,j,k}+(Theta_X)_{i+1,j,k}}{h_x^2} + $

$latex frac{(Theta_X)_{i,j-1,k}-2(Theta_X)_{i,j,k}+(Theta_X)_{i,j+1,k}}{h_y^2} + $

$latex frac{(Theta_X)_{i,j,k-1}-2(Theta_X)_{i,j,k}+(Theta_X)_{i,j,k+1}}{h_z^2} = $

$latex = 6 pi (partial_x S^*_x + partial_y S^*_y +partial_z S^*_z )_{i,j,k} $,

donde inicialmente:

$latex (S^*_a)_{i,j,k} = (psi^6)_{i,j,k}rho_{i,j,k}h_{i,j,k}w^2_{i,j,k}(v_a)_{i,j,k}$,

$latex (partial_x S^*_x + partial_y S^*_y +partial_z S^*_z )_{i,j,k} = $

$latex frac{(S^*_x)_{i+1,j,k}-(S^*_x)_{i-1,j,k}}{2h_x} + frac{(S^*_x)_{i,j+1,k}-(S^*_x)_{i,j-1,k}}{2h_y} + frac{(S^*_x)_{i,j,k+1}-(S^*_x)_{i,j,k-1}}{2h_z}$

y que es lineal.

El primer sistema acoplado de ecuaciones quedaría ahora:

$latex partial_{xx} X^x + partial_{yy} X^x + partial_{zz} X^x = 8 pi S^*_x – frac{1}{3} partial_x Theta_X approx $

$latex approx frac{X^x_{i-1,j,k}-2X^x_{i,j,k}+X^x_{i+1,j,k}}{h_x^2} + frac{X^x_{i,j-1,k}-2X^x_{i,j,k}+X^x_{i,j+1,k}}{h_y^2} + frac{X^x_{i,j,k-1}-2X^x_{i,j,k}+X^x_{i,j,k+1}}{h_z^2} = $

$latex = 8 pi (S^*_x)_{i,j,k} – frac{1}{3} (partial_x Theta_X)_{i,j,k}$,

¡que vuelve a ser lineal!

Continuamos con:

$latex partial_{xx} X^y + partial_{yy} X^y + partial_{zz} X^y = 8 pi S^*_y – frac{1}{3} partial_y Theta_X approx $

$latex approx frac{X^y_{i-1,j,k}-2X^y_{i,j,k}+X^y_{i+1,j,k}}{h_x^2} + frac{X^y_{i,j-1,k}-2X^y_{i,j,k}+X^y_{i,j+1,k}}{h_y^2} + frac{X^y_{i,j,k-1}-2X^y_{i,j,k}+X^y_{i,j,k+1}}{h_z^2} = $

$latex = 8 pi (S^*_y)_{i,j,k} – frac{1}{3} (partial_y Theta_X)_{i,j,k}$

y, finalmente:

$latex partial_{xx} X^z + partial_{yy} X^z + partial_{zz} X^z = 8 pi S^*_z – frac{1}{3} partial_z Theta_X approx $

$latex approx frac{X^z_{i-1,j,k}-2X^z_{i,j,k}+X^z_{i+1,j,k}}{h_x^2} + frac{X^z_{i,j-1,k}-2X^z_{i,j,k}+X^z_{i,j+1,k}}{h_y^2} + frac{X^z_{i,j,k-1}-2X^z_{i,j,k}+X^z_{i,j,k+1}}{h_z^2} = $

$latex = 8 pi (S^*_z)_{i,j,k} – frac{1}{3} (partial_z Theta_X)_{i,j,k}$,

donde calculamos al principio:

$latex (partial_x Theta_X)_{i,j,k} = frac{(Theta_X)_{i+1,j,k}-(Theta_X)_{i-1,j,k}}{2h_x}$

$latex (partial_y Theta_X)_{i,j,k} = frac{(Theta_X)_{i,j+1,k}-(Theta_X)_{i,j-1,k}}{2h_y}$

$latex (partial_z Theta_X)_{i,j,k} = frac{(Theta_X)_{i,j,k+1} – (Theta_X)_{i,j,k-1}}{2h_z}$

A continuación, discretizamos las siguientes ecuaciones:

$latex hat{A}^{xx} = 2 partial_x X^x – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z) approx$

$latex approx frac{2}{3}frac{X^x_{i+1,j,k}-X^x_{i-1,j,k}}{h_x} -frac{1}{3} frac{X^y_{i,j+1,k}-X^y_{i,j-1,k}}{h_y} – frac{1}{3} frac{X^z_{i,j,k+1}-X^z_{i,j,k-1}}{h_z}) = hat{A}^{xx}_{i,j,k}$,

$latex hat{A}^{xy} = hat{A}^{yx}= partial_x X^y + partial_y X^x approx $

$latex approx frac{X^y_{i+1,j,k}-X^y_{i-1,j,k}}{2h_x} + frac{X^x_{i,j+1,k}-X^x_{i,j-1,k}}{2h_y} = hat{A}^{xy}_{i,j,k} = hat{A}^{yx}_{i,j,k}$,

$latex hat{A}^{xz} = hat{A}^{zx} = partial_x X^z + partial_z X^x approx $

$latex approx frac{X^z_{i+1,j,k}-X^z_{i-1,j,k}}{2h_x} + frac{X^x_{i,j,k+1}-X^x_{i,j,k-1}}{2h_z} = hat{A}^{xz}_{i,j,k} = hat{A}^{zx}_{i,j,k}$,

$latex hat{A}^{yy} = 2 partial_y X^y – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z) approx $

$latex approx -frac{1}{3}frac{X^x_{i+1,j,k}-X^x_{i-1,j,k}}{h_x} +frac{2}{3} frac{X^y_{i,j+1,k}-X^y_{i,j-1,k}}{h_y} – frac{1}{3} frac{X^z_{i,j,k+1}-X^z_{i,j,k-1}}{h_z}) = hat{A}^{yy}_{i,j,k}$,

$latex hat{A}^{yz} = hat{A}^{zy} = partial_y X^z + partial_z X^y approx $

$latex approx frac{X^z_{i,j+1,k}-X^z_{i,j-1,k}}{2h_y} + frac{X^y_{i,j,k+1}-X^y_{i,j,k-1}}{2h_z} = hat{A}^{yz}_{i,j,k} = hat{A}^{zy}_{i,j,k}$,

$latex hat{A}^{zz} = 2 partial_z X^z – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z) approx $

$latex approx -frac{1}{3}frac{X^x_{i+1,j,k}-X^x_{i-1,j,k}}{h_x} -frac{1}{3} frac{X^y_{i,j+1,k}-X^y_{i,j-1,k}}{h_y} + frac{2}{3} frac{X^z_{i,j,k+1}-X^z_{i,j,k-1}}{h_z}) = hat{A}^{zz}_{i,j,k}$.

Por tanto, la siguiente ecuación:

$latex Delta psi = -2 pi psi^{-1} E^* – psi^{-7} frac{(hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2}{8}$

queda:

$latex approx frac{psi_{i-1,j,k}-2psi_{i,j,k}+psi_{i+1,j,k}}{h_x^2} + frac{psi_{i,j-1,k}-2psi_{i,j,k}+psi_{i,j+1,k}}{h_y^2} + frac{psi_{i,j,k-1}-2psi_{i,j,k}+psi_{i,j,k+1}}{h_z^2} = $

$latex =-2 pi psi^{-1}_{i,j,k} E^*_{i,j,k} – $

$latex – frac{psi^{-7}_{i,j,k}}{8} ( (hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2 ) $,

con:

$latex partial_{psi_{i,j,k}} F(psi_{i,j,k}) = -2 ( frac{1}{h_x^2} + frac{1}{h_y^2} + frac{1}{h_z^2} ) -2 pi psi_{i,j,k}^{-2} E^*_{i,j,k} – $

$latex – frac{7}{8} psi^{-8}_{i,j,k} ( (hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2 )$,

donde:

$latex E^*_{i,j,k} = psi^{6}_{i,j,k} (D_{i,j,k}+tau_{i,j,k})$

y la ecuación:

$latex Delta (alphapsi) = (alpha psi) (2 pi psi^{-2} (E^*+2S^*) + $

$latex + frac{7}{8} psi^{-8} ((hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2) )$

como:

$latex approx frac{(alphapsi)_{i-1,j,k} – 2(alphapsi)_{i,j,k}+(alphapsi)_{i+1,j,k}}{h_x^2} + $

$latex + frac{(alphapsi)_{i,j-1,k}-2(alphapsi)_{i,j,k}+(alphapsi)_{i,j+1,k}}{h_y^2} + $

$latex + frac{(alphapsi)_{i,j,k-1}-2(alphapsi)_{i,j,k}+(alphapsi)_{i,j,k+1}}{h_z^2} = $

$latex = (alpha psi)_{i,j,k} (2 pi psi^{-2}_{i,j,k} (E^*_{i,j,k}+2S^*_{i,j,k}) + $

$latex + frac{7}{8} psi^{-8}_{i,j,k} ((hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2) )$,

donde:

$latex partial_{(alpha psi)_{i,j,k}} F((alpha psi)_{i,j,k}) = -2 ( frac{1}{h_x^2} + frac{1}{h_y^2} + frac{1}{h_z^2} ) – 2 pi psi^{-2}_{i,j,k} (E^*_{i,j,k}+2S^*_{i,j,k}) + $

$latex – frac{7}{8} psi^{-8}_{i,j,k} ((hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2) )$

con:

$latex S^*_{i,j,k} = psi^6_{i,j,k}(rho_{i,j,k}h_{i,j,k}(w^2_{i,j,k}-1) + 3 p_{i,j,k})$.

Finalmente, tenemos el otro sistema acoplado:

$latex Delta beta^i = mathcal{D}_j(2 alpha psi^{-6} hat{A}^{ij}) – frac{1}{3} mathcal{D}^i(mathcal{D}_j beta^j)$,

con el que procedemos de igual manera que con las $latex X^i$:

$latex Delta(mathcal{D}_i beta^i) = mathcal{D}_i (mathcal{D}_j (2 alpha psi^{-6} hat{A}^{ij})) – frac{1}{3} Delta (mathcal{D}_i beta^i)$,

de manera que:

$latex Delta Theta_beta = frac{3}{4} mathcal{D}^i (mathcal{D}_j (2 alpha psi^{-6} hat{A}^{ij})) =$

$latex frac{3}{2}(partial_{xx}(alpha psi^{-6} hat{A}^{xx}) + partial_{yy}(alpha psi^{-6} hat{A}^{yy}) + partial_{zz}(alpha psi^{-6} hat{A}^{zz})$,

con:

$latex Theta_beta := mathcal{D}_i beta^i$,

que discretizada queda:

$latex frac{(Theta_beta)_{i-1,j,k}-2(Theta_beta)_{i,j,k}+(Theta_beta)_{i+1,j,k}}{h_x^2} + $

$latex frac{(Theta_beta)_{i,j-1,k}-2(Theta_beta)_{i,j,k}+(Theta_beta)_{i,j+1,k}}{h_y^2} + $

$latex frac{(Theta_beta)_{i,j,k-1}-2(Theta_beta)_{i,j,k}+(Theta_beta)_{i,j,k+1}}{h_z^2} = $

$latex frac{3}{2}((partial_{xx}(alpha psi^{-6} hat{A}^{xx}))_{i,j,k} + (partial_{yy}(alpha psi^{-6} hat{A}^{yy}))_{i,j,k} + (partial_{zz}(alpha psi^{-6} hat{A}^{zz})_{i,j,k})$,

De esta manera, tenemos:

$latex Delta beta^x = partial_x (2 alpha psi^{-6} hat{A}^{xx}) + partial_y (2 alpha psi^{-6} hat{A}^{xy}) + partial_z (2 alpha psi^{-6} hat{A}^{xz}) – frac{1}{2} partial_x Theta_beta approx $

$latex approx frac{beta^x_{i-1,j,k}-2beta^x_{i,j,k}+beta^x_{i+1,j,k}}{h_x^2} + frac{beta^x_{i,j-1,k}-2beta^x_{i,j,k}+beta^x_{i,j+1,k}}{h_y^2} + frac{beta^x_{i,j,k-1}-2beta^x_{i,j,k}+beta^x_{i,j,k+1}}{h_z^2} = $

$latex = (partial_x (2 alpha psi^{-6} hat{A}^{xx}))_{i,j,k} + (partial_y (2 alpha psi^{-6} hat{A}^{xy}))_{i,j,k} + (partial_z (2 alpha psi^{-6} hat{A}^{xz}) )_{i,j,k} – $

$latex – frac{1}{3} (partial_x Theta_beta)_{i,j,k}$.

De la misma manera:

$latex Delta beta^y = partial_x (2 alpha psi^{-6} hat{A}^{yx}) + partial_y (2 alpha psi^{-6} hat{A}^{yy}) + partial_z (2 alpha psi^{-6} hat{A}^{yz}) – frac{1}{3} partial_y Theta_beta approx $

$latex approx frac{beta^y_{i-1,j,k}-2beta^y_{i,j,k}+beta^y_{i+1,j,k}}{h_x^2} + frac{beta^y_{i,j-1,k}-2beta^y_{i,j,k}+beta^y_{i,j+1,k}}{h_y^2} + frac{beta^y_{i,j,k-1}-2beta^y_{i,j,k}+beta^y_{i,j,k+1}}{h_z^2} = $

$latex = (partial_x (2 alpha psi^{-6} hat{A}^{yx}))_{i,j,k} + (partial_y (2 alpha psi^{-6} hat{A}^{yy}))_{i,j,k} + (partial_z (2 alpha psi^{-6} hat{A}^{yz}) )_{i,j,k} – $

$latex – frac{1}{3} (partial_y Theta_beta)_{i,j,k}$.

Y, por último:

$latex Delta beta^z = partial_x (2 alpha psi^{-6} hat{A}^{zx}) + partial_y (2 alpha psi^{-6} hat{A}^{zy}) + partial_z (2 alpha psi^{-6} hat{A}^{zz}) – frac{1}{3} partial_z Theta_beta approx $

$latex approx frac{beta^z_{i-1,j,k}-2beta^z_{i,j,k}+beta^z_{i+1,j,k}}{h_x^2} + frac{beta^z_{i,j-1,k}-2beta^z_{i,j,k}+beta^z_{i,j+1,k}}{h_y^2} + frac{beta^z_{i,j,k-1}-2beta^z_{i,j,k}+beta^z_{i,j,k+1}}{h_z^2} = $

$latex = (partial_x (2 alpha psi^{-6} hat{A}^{zx}))_{i,j,k} + (partial_y (2 alpha psi^{-6} hat{A}^{zy}))_{i,j,k} + (partial_z (2 alpha psi^{-6} hat{A}^{zz}) )_{i,j,k} – $

$latex – frac{1}{3} (partial_z Theta_beta)_{i,j,k}$.

Parece que, del sistema no lineal acoplado inicial, hemos llegado a un sistema de diez ecuaciones desacopladas donde ocho de ellas son lineales y solo dos son no linales. No pinta mal. Ya escribiremos próximamente sobre las condiciones de contorno…

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Vamos a discretizar las ecuaciones que comentamos en este post. Para ello, discretizaremos las derivadas de la siguiente manera:

$latex partial_x u = frac{u_{i+1,j,k}-u_{i-1,j,k}}{2h_x}$,

$latex partial_y u = frac{u_{i,j+1,k}-u_{i,j-1,k}}{2h_y}$,

$latex partial_z u = frac{u_{i,j,k+1}-u_{i,j,k-1}}{2h_z}$,

$latex partial_{xx} u = frac{u_{i-1,j,k}-2u_{i,j,k}+u_{i+1,j,k}}{h_x^2}$,

$latex partial_{yy} u = frac{u_{i,j-1,k}-2u_{i,j,k}+u_{i,j+1,k}}{h_y^2}$,

$latex partial_{zz} u = frac{u_{i,j,k-1}-2u_{i,j,k}+u_{i,j,k+1}}{h_z^2}$,

$latex partial_{xy} u = frac{u_{i-1,j-1,k}-u_{i+1,j-1,k}-u_{i-1,j+1,k}+u_{i+1,j+1,k}}{4h_xh_y}$,

$latex partial_{xz} u = frac{u_{i-1,j,k-1}-u_{i+1,j,k-1}-u_{i-1,j,k+1}+u_{i+1,j,k+1}}{4h_xh_z}$,

$latex partial_{yz} u = frac{u_{i,j-1,k-1}-u_{i,j+1,k-1}-u_{i,j-1,k+1}+u_{i,j+1,k+1}}{4h_yh_z}$.

El primer grupo de ecuaciones quedaría:

$latex partial_{xx} X^x + partial_{yy} X^x + partial_{zz} X^x = 8 pi psi^6 rho h w^2 v_x – frac{1}{3} partial_x (partial_x X^x + partial_y X^y + partial_z X^z) approx $

$latex approx frac{X^x_{i-1,j,k}-2X^x_{i,j,k}+X^x_{i+1,j,k}}{h_x^2} + frac{X^x_{i,j-1,k}-2X^x_{i,j,k}+X^x_{i,j+1,k}}{h_y^2} + frac{X^x_{i,j,k-1}-2X^x_{i,j,k}+X^x_{i,j,k+1}}{h_z^2} = $

$latex = 8 pi psi^6_{i,j,k} rho_{i,j,k} h_{i,j,k} w^2_{i,j,k} v_{x_{i,j,k}} – frac{1}{3} ( frac{X^x_{i-1,j,k}-2X^x_{i,j,k}+X^x_{i+1,j,k}}{h_x^2} +$

$latex + frac{X^y_{i-1,j-1,k}-X^y_{i+1,j-1,k}-X^y_{i-1,j+1,k}+X^y_{i+1,j+1,k}}{4h_xh_y} +$

$latex + frac{X^z_{i-1,j,k-1}-X^z_{i+1,j,k-1}-X^z_{i-1,j,k+1}+X^z_{i+1,j,k+1}}{4h_xh_z} )$,

y además, para los esquemas de relajación no lineales, reescribimos la igualdad anterior como $latex F(X^x_{i,j,k})=0$ y entonces tenemos:

$latex partial_{X^x_{i,j,k}} F(X^x_{i,j,k}) = -2 ( frac{4}{3}frac{1}{h_x^2} + frac{1}{h_y^2} + frac{1}{h_z^2})$.

$latex partial_{xx} X^y + partial_{yy} X^y + partial_{zz} X^y = 8 pi psi^6 rho h w^2 v_y – frac{1}{3} partial_y (partial_x X^x + partial_y X^y + partial_z X^z) approx $

$latex approx frac{X^y_{i-1,j,k}-2X^y_{i,j,k}+X^y_{i+1,j,k}}{h_x^2} + frac{X^y_{i,j-1,k}-2X^y_{i,j,k}+X^y_{i,j+1,k}}{h_y^2} + frac{X^y_{i,j,k-1}-2X^y_{i,j,k}+X^y_{i,j,k+1}}{h_z^2} = $

$latex = 8 pi psi^6_{i,j,k} rho_{i,j,k} h_{i,j,k} w^2_{i,j,k} v_{y_{i,j,k}} – frac{1}{3} ( frac{X^x_{i-1,j-1,k}-X^x_{i+1,j-1,k}-X^x_{i-1,j+1,k}+X^x_{i+1,j+1,k}}{4h_xh_y} +$

$latex + frac{X^y_{i,j-1,k}-2X^y_{i,j,k}+X^y_{i,j+1,k}}{h_y^2} +$

$latex + frac{X^z_{i-1,j,k-1}-X^z_{i+1,j,k-1}-X^z_{i-1,j,k+1}+X^z_{i+1,j,k+1}}{4h_yh_z} )$,

con:

$latex partial_{X^y_{i,j,k}} F(X^y_{i,j,k}) = -2 ( frac{1}{h_x^2} +frac{4}{3} frac{1}{h_y^2} + frac{1}{h_z^2})$.

$latex partial_{xx} X^z + partial_{yy} X^z + partial_{zz} X^z = 8 pi psi^6 rho h w^2 v_z – frac{1}{3} partial_z (partial_x X^x + partial_y X^y + partial_z X^z) approx $

$latex approx frac{X^z_{i-1,j,k}-2X^z_{i,j,k}+X^z_{i+1,j,k}}{h_x^2} + frac{X^z_{i,j-1,k}-2X^z_{i,j,k}+X^z_{i,j+1,k}}{h_y^2} + frac{X^z_{i,j,k-1}-2X^z_{i,j,k}+X^z_{i,j,k+1}}{h_z^2} = $

$latex = 8 pi psi^6_{i,j,k} rho_{i,j,k} h_{i,j,k} w^2_{i,j,k} v_{z_{i,j,k}} – frac{1}{3} ( frac{X^x_{i-1,j,k-1}-X^x_{i+1,j,k-1}-X^x_{i-1,j,k+1}+X^x_{i+1,j,k+1}}{4h_xh_z} +$

$latex + frac{X^y_{i,j-1,k-1}-X^y_{i,j+1,k-1}-X^y_{i,j-1,k+1}+X^y_{i,j+1,k+1}}{4h_yh_z} )$

$latex + frac{X^z_{i,j,k-1}-2X^z_{i,j,k}+X^z_{i,j,k+1}}{h_z^2}$

con:

$latex partial_{X^z_{i,j,k}} = F(X^z_{i,j,k}) = -2 ( frac{1}{h_x^2} + frac{1}{h_y^2} + frac{4}{3} frac{1}{h_z^2})$.

A continuación, discretizamos las siguientes ecuaciones:

$latex hat{A}^{xx} = 2 partial_x X^x – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z) approx$

$latex approx frac{2}{3}frac{X^x_{i+1,j,k}-X^x_{i-1,j,k}}{h_x} -frac{1}{3} frac{X^y_{i,j+1,k}-X^y_{i,j-1,k}}{h_y} – frac{1}{3} frac{X^z_{i,j,k+1}-X^z_{i,j,k-1}}{h_z}) = hat{A}^{xx}_{i,j,k}$,

$latex hat{A}^{xy} = hat{A}^{yx}= partial_x X^y + partial_y X^x approx $

$latex approx frac{X^y_{i+1,j,k}-X^y_{i-1,j,k}}{2h_x} + frac{X^x_{i,j+1,k}-X^x_{i,j-1,k}}{2h_y} = hat{A}^{xy}_{i,j,k} = hat{A}^{yx}_{i,j,k}$,

$latex hat{A}^{xz} = hat{A}^{zx} = partial_x X^z + partial_z X^x approx $

$latex approx frac{X^z_{i+1,j,k}-X^z_{i-1,j,k}}{2h_x} + frac{X^x_{i,j,k+1}-X^x_{i,j,k-1}}{2h_z} = hat{A}^{xz}_{i,j,k} = hat{A}^{zx}_{i,j,k}$,

$latex hat{A}^{yy} = 2 partial_y X^y – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z) approx $

$latex approx -frac{1}{3}frac{X^x_{i+1,j,k}-X^x_{i-1,j,k}}{h_x} +frac{2}{3} frac{X^y_{i,j+1,k}-X^y_{i,j-1,k}}{h_y} – frac{1}{3} frac{X^z_{i,j,k+1}-X^z_{i,j,k-1}}{h_z}) = hat{A}^{yy}_{i,j,k}$,

$latex hat{A}^{yz} = hat{A}^{zy} = partial_y X^z + partial_z X^y approx $

$latex approx frac{X^z_{i,j+1,k}-X^z_{i,j-1,k}}{2h_y} + frac{X^y_{i,j,k+1}-X^y_{i,j,k-1}}{2h_z} = hat{A}^{yz}_{i,j,k} = hat{A}^{zy}_{i,j,k}$,

$latex hat{A}^{zz} = 2 partial_z X^z – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z) approx $

$latex approx -frac{1}{3}frac{X^x_{i+1,j,k}-X^x_{i-1,j,k}}{h_x} -frac{1}{3} frac{X^y_{i,j+1,k}-X^y_{i,j-1,k}}{h_y} + frac{2}{3} frac{X^z_{i,j,k+1}-X^z_{i,j,k-1}}{h_z}) = hat{A}^{zz}_{i,j,k}$.

Por tanto, la siguiente ecuación:

$latex Delta psi = -2 pi psi^{-1} (D + tau) – psi^{-7} frac{(hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2}{8}$

queda:

$latex approx frac{psi_{i-1,j,k}-2psi_{i,j,k}+psi_{i+1,j,k}}{h_x^2} + frac{psi_{i,j-1,k}-2psi_{i,j,k}+psi_{i,j+1,k}}{h_y^2} + frac{psi_{i,j,k-1}-2psi_{i,j,k}+psi_{i,j,k+1}}{h_z^2} = $

$latex =-2 pi psi^{-1}_{i,j,k} (D_{i,j,k}+tau_{i,j,k}) – $

$latex – frac{psi^{-7}_{i,j,k}}{8} ( (hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2 ) $,

con:

$latex partial_{psi_{i,j,k}} F(psi_{i,j,k}) = -2 ( frac{1}{h_x^2} + frac{1}{h_y^2} + frac{1}{h_z^2} ) -2 pi psi_{i,j,k}^{-2} (D_{i,j,k}+tau_{i,j,k}) – $

$latex – frac{7}{8} psi^{-8}_{i,j,k} ( (hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2 )$.

y la ecuación:

$latex Delta (alphapsi) = 2 pi (alphapsi)^{-1} ( D + tau + 2 rho h (w^2-1) + 6 p) + $

$latex + frac{7}{8} (alpha psi)^{-7} ((hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2)$

como:

$latex approx frac{(alphapsi)_{i-1,j,k} – 2(alphapsi)_{i,j,k}+(alphapsi)_{i+1,j,k}}{h_x^2} + frac{(alphapsi)_{i,j-1,k}-2(alphapsi)_{i,j,k}+(alphapsi)_{i,j+1,k}}{h_y^2} + frac{(alphapsi)_{i,j,k-1}-2(alphapsi)_{i,j,k}+(alphapsi)_{i,j,k+1}}{h_z^2} = $

$latex =2 pi (alphapsi)_{i,j,k}^{-1} (D_{i,j,k}+tau_{i,j,k} + 2 rho_{i,j,k} h_{i,j,k} (w^2_{i,j,k}-1)+6p_{i,j,k}) + $

$latex + frac{7}{8}(alphapsi)_{i,j,k}^{-7} ( (hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2 ) $,

donde:

$latex partial_{psialpha_{i,j,k}} F(psialpha_{i,j,k}) = -2 ( frac{1}{h_x^2} + frac{1}{h_y^2} + frac{1}{h_z^2} ) + $

$latex + 2 pi (psialpha)_{i,j,k}^{-2} (D_{i,j,k}+tau_{i,j,k} + 2 rho_{i,j,k} h_{i,j,k} (w^2_{i,j,k}-1)+6p_{i,j,k}) – $

$latex + frac{49}{8} (psialpha)_{i,j,k}^{-8} ( (hat{A}^{xx}_{i,j,k})^2+(hat{A}^{yy}_{i,j,k})^2+(hat{A}^{zz}_{i,j,k})^2+2(hat{A}^{xy}_{i,j,k})^2+2(hat{A}^{xz}_{i,j,k})^2+2(hat{A}^{yz}_{i,j,k})^2 )$.

Finalmente, tenemos:

$latex Delta beta^x = partial_x (2 alpha psi^{-6} hat{A}^{xx}) + partial_y (2 alpha psi^{-6} hat{A}^{xy}) + partial_z (2 alpha psi^{-6} hat{A}^{xz}) – $

$latex – frac{1}{3} partial_x (partial_x beta^x + partial_y beta^y + partial_z beta^z) approx $

$latex approx frac{beta^x_{i-1,j,k}-2beta^x_{i,j,k}+beta^x_{i+1,j,k}}{h_x^2} + frac{beta^x_{i,j-1,k}-2beta^x_{i,j,k}+beta^x_{i,j+1,k}}{h_y^2} + frac{beta^x_{i,j,k-1}-2beta^x_{i,j,k}+beta^x_{i,j,k+1}}{h_z^2} = $

$latex = frac{(alpha psi)_{i+1,j,k}^{-6} hat{A}_{i+1,j,k}^{xx} – (alpha psi)_{i-1,j,k}^{-6} hat{A}_{i-1,j,k}^{xx}}{h_x} + $

$latex + frac{(alpha psi)_{i,j+1,k}^{-6} hat{A}_{i,j+1,k}^{xy} – (alpha psi)_{i,j-1,k}^{-6} hat{A}_{i,j-1,k}^{xy}}{h_y} + $

$latex + frac{(alpha psi)_{i,j,k+1}^{-6} hat{A}_{i,j,k+1}^{xz} – (alpha psi)_{i,j,k-1}^{-6} hat{A}_{i,j,k-1}^{xz}}{h_z} – $

$latex – frac{1}{3} ( frac{beta^x_{i-1,j,k}-2beta^x_{i,j,k}+beta^x_{i+1,j,k}}{h_x^2} + $

$latex + frac{beta^y_{i-1,j-1,k}-beta^y_{i+1,j-1,k}-beta^y_{i-1,j+1,k}+beta^y_{i+1,j+1,k}}{4 h_x h_y} + $

$latex + frac{beta^z_{i-1,j,k-1}-beta^z_{i+1,j,k-1}-beta^z_{i-1,j,k+1}+beta^z_{i+1,j,k+1}}{4 h_x h_z} $,

con:

$latex partial_{beta^x_{i,j,k}} F(beta^x_{i,j,k}) = -2 ( frac{4}{3}frac{1}{h_x^2} + frac{1}{h_y^2} + frac{1}{h_z^2})$,

$latex Delta beta^y = partial_x (2 alpha psi^{-6} hat{A}^{yx}) + partial_y (2 alpha psi^{-6} hat{A}^{yy}) + partial_z (2 alpha psi^{-6} hat{A}^{yz}) – $

$latex – frac{1}{3} partial_y (partial_x beta^x + partial_y beta^y + partial_z beta^z) approx $

$latex approx frac{beta^y_{i-1,j,k}-2beta^y_{i,j,k}+beta^y_{i+1,j,k}}{h_x^2} + frac{beta^y_{i,j-1,k}-2beta^y_{i,j,k}+beta^y_{i,j+1,k}}{h_y^2} + frac{beta^y_{i,j,k-1}-2beta^y_{i,j,k}+beta^y_{i,j,k+1}}{h_z^2} = $

$latex = frac{(alpha psi)_{i+1,j,k}^{-6} hat{A}_{i+1,j,k}^{yx} – (alpha psi)_{i-1,j,k}^{-6} hat{A}_{i-1,j,k}^{yx}}{h_x} + $

$latex + frac{(alpha psi)_{i,j+1,k}^{-6} hat{A}_{i,j+1,k}^{yy} – (alpha psi)_{i,j-1,k}^{-6} hat{A}_{i,j-1,k}^{yy}}{h_y} + $

$latex + frac{(alpha psi)_{i,j,k+1}^{-6} hat{A}_{i,j,k+1}^{yz} – (alpha psi)_{i,j,k-1}^{-6} hat{A}_{i,j,k-1}^{yz}}{h_z} – $

$latex – frac{1}{3} ( frac{beta^x_{i-1,j-1,k}-beta^x_{i+1,j-1,k}-beta^x_{i-1,j+1,k}+beta^x_{i+1,j+1,k}}{4h_xh_y} +$

$latex + frac{beta^y_{i,j-1,k}-2beta^y_{i,j,k}+beta^y_{i,j+1,k}}{h_y^2} +$

$latex + frac{beta^z_{i-1,j,k-1}-beta^z_{i+1,j,k-1}-beta^z_{i-1,j,k+1}+beta^z_{i+1,j,k+1}}{4h_yh_z} )$,

con:

$latex partial_{beta^y_{i,j,k}} F(beta^y_{i,j,k}) = -2 ( frac{1}{h_x^2} + frac{4}{3} frac{1}{h_y^2} + frac{1}{h_z^2})$,

$latex Delta beta^z = partial_x (2 alpha psi^{-6} hat{A}^{zx}) + partial_y (2 alpha psi^{-6} hat{A}^{zy}) + partial_z (2 alpha psi^{-6} hat{A}^{zz}) – $

$latex – frac{1}{3} partial_z (partial_x beta^x + partial_y beta^y + partial_z beta^z) approx $

$latex approx frac{beta^z_{i-1,j,k}-2beta^z_{i,j,k}+beta^z_{i+1,j,k}}{h_x^2} + frac{beta^z_{i,j-1,k}-2beta^z_{i,j,k}+beta^z_{i,j+1,k}}{h_y^2} + frac{beta^z_{i,j,k-1}-2beta^z_{i,j,k}+beta^z_{i,j,k+1}}{h_z^2} = $

$latex = frac{(alpha psi)_{i+1,j,k}^{-6} hat{A}_{i+1,j,k}^{zx} – (alpha psi)_{i-1,j,k}^{-6} hat{A}_{i-1,j,k}^{zx}}{h_x} + $

$latex + frac{(alpha psi)_{i,j+1,k}^{-6} hat{A}_{i,j+1,k}^{zy} – (alpha psi)_{i,j-1,k}^{-6} hat{A}_{i,j-1,k}^{zy}}{h_y} + $

$latex + frac{(alpha psi)_{i,j,k+1}^{-6} hat{A}_{i,j,k+1}^{zz} – (alpha psi)_{i,j,k-1}^{-6} hat{A}_{i,j,k-1}^{zz}}{h_z} – $

$latex – frac{1}{3} ( frac{beta^x_{i-1,j,k-1}-beta^x_{i+1,j,k-1}-beta^x_{i-1,j,k+1}+beta^x_{i+1,j,k+1}}{4h_xh_z} +$

$latex + frac{beta^y_{i,j-1,k-1}-beta^y_{i,j+1,k-1}-beta^y_{i,j-1,k+1}+beta^y_{i,j+1,k+1}}{4h_yh_z} )$

$latex + frac{beta^z_{i,j,k-1}-2beta^z_{i,j,k}+beta^z_{i,j,k+1}}{h_z^2}$,

con:

$latex partial_{beta^z_{i,j,k}} F(beta^z_{i,j,k}) = -2 ( frac{1}{h_x^2} + frac{1}{h_y^2} + frac{4}{3} frac{1}{h_z^2} )$.

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Ya escribimos al respecto en este post. Aquí lo que haremos es reescribir las expresiones allí introducidas

En primer lugar, teniamos:

 $latex Delta X^i = 8 pi f^{ij}S_j^* – frac{1}{3}mathcal{D}^i mathcal{D}_j X^j$

donde:

$latex S_j^* := sqrt{ frac{gamma}{f} } S = psi^6 S_j$,

$latex S_j := rho h w^2 v_j$.

En el caso de estar trabajando en cartesianas y teniendo en cuenta todo el trabajo realizado en el artículo, nos queda:

$latex partial_{xx} X^x + partial_{yy} X^x + partial_{zz} X^x = 8 pi psi^6 rho h w^2 v_x – frac{1}{3} partial_x (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex partial_{xx} X^y + partial_{yy} X^y + partial_{zz} X^y = 8 pi psi^6 rho h w^2 v_y – frac{1}{3} partial_y (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex partial_{xx} X^z + partial_{yy} X^z + partial_{zz} X^z = 8 pi psi^6 rho h w^2 v_z – frac{1}{3} partial_z (partial_x X^x + partial_y X^y + partial_z X^z)$.

A continuación, y para la siguiente ecuación, necesitamos:

$latex hat{A}^{ij} = mathcal{D}^i X^j + mathcal{D}^j X^i – frac{2}{3} mathcal{D}_k X^k f^{ij}$

que queda como:

$latex hat{A}^{xx} = 2 partial_x X^x – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex hat{A}^{xy} = hat{A}^{yx}= partial_x X^y + partial_y X^x$,

$latex hat{A}^{xz} = hat{A}^{zx} = partial_x X^z + partial_z X^x$,

$latex hat{A}^{yy} = 2 partial_y X^y – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex hat{A}^{yz} = hat{A}^{zy} = partial_y X^z + partial_z X^y$,

$latex hat{A}^{zz} = 2 partial_z X^z – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z)$,

por lo que:

$latex Delta psi = -2 pi psi^{-1} E^* – psi^{-7} frac{f_{il}f_{jm}hat{A}^{lm}hat{A}^{ij}}{8}$

donde:

$latex E^*:= sqrt{ frac{gamma}{f} } E = psi^6 E$,

$latex E:= D + tau$

es:

$latex Delta psi = -2 pi psi^{-1} (D + tau) – psi^{-7} frac{(hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2}{8}$.

La siguiente:

$latex Delta (alphapsi) = 2 pi (alphapsi)^{-1} (E^* + 2S^*) + frac{7}{8} (alphapsi)^{-7} (f_{il} f{jm} hat{A}^{lm} hat{A}^{ij})$

con:

$latex S^*:= sqrt{ frac{gamma}{f} } S = psi^6 S$,

$latex S:= rho h (w^2-1) + 3 p$

queda:

$latex Delta (alphapsi) = 2 pi (alphapsi)^{-1} ( D + tau + 2 rho h (w^2-1) + 6 p) + $

$latex + frac{7}{8}(alphapsi)^{-7} ((hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2)$

Y la última:

$latex Delta beta^i = mathcal{D}_j (2 (alphapsi)^{-6} hat{A}^{ij}) – frac{1}{3} mathcal{D}^i (mathcal{D}_j beta^j)$,

que escribimos como:

$latex Delta beta^x = partial_x (2 (alpha psi)^{-6} hat{A}^{xx}) + partial_y (2 (alpha psi)^{-6} hat{A}^{xy}) + partial_z (2 (alpha psi)^{-6} hat{A}^{xz}) – $

$latex – frac{1}{3} partial_x (partial_x beta^x + partial_y beta^y + partial_z beta^z)$

$latex Delta beta^y = partial_x (2 (alpha psi)^{-6} hat{A}^{yx}) + partial_y (2 (alpha psi)^{-6} hat{A}^{yy}) + partial_z (2 (alpha psi)^{-6} hat{A}^{yz}) – $

$latex – frac{1}{3} partial_y (partial_x beta^x + partial_y beta^y + partial_z beta^z)$

$latex Delta beta^z = partial_x (2 (alpha psi)^{-6} hat{A}^{zx}) + partial_y (2 (alpha psi)^{-6} hat{A}^{zy}) + partial_z (2 (alpha psi)^{-6} hat{A}^{zz}) – $

$latex – frac{1}{3} partial_z (partial_x beta^x + partial_y beta^y + partial_z beta^z)$

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CoCoNuT es un código que permite realizar simulaciones de colapso estelar. Reescribimos las ecuaciones CFC, que son un caso particular de la aproximación FCF haciendo que las $latex h^{ij}$ sean cero, en terminos de las variables que éste utiliza. Empezamos con una auxilar:

 $latex Delta X^i = 8 pi f^{ij}S_j^* – frac{1}{3}mathcal{D}^i mathcal{D}_j X^j$

donde:

$latex S_j^* := sqrt{ frac{gamma}{f} } S = psi^6 S_j$,

$latex S_j := rho h w^2 v_j$.

La primera es:

$latex Delta psi = -2 pi psi^{-1} E^* – psi^{-7} frac{f_{il}f_{jm}hat{A}^{lm}hat{A}^{ij}}{8}$

donde:

$latex E^*:= sqrt{ frac{gamma}{f} } E = psi^6 E$,

$latex E:= D + tau$

La siguiente:

$latex Delta (psi alpha) = 2 pi alpha (E^* + 2S^*) + alpha psi^{-7} frac{7 f_{il} f{jm} hat{A}^{lm} hat{A}^{ij}}{8}$

con:

$latex S^*:= sqrt{ frac{gamma}{f} } S = psi^6 S$,

$latex S:= rho h (w^2-1) + 3 p$

Y la última:

$latex Delta beta^i = mathcal{D}_j (2 alpha psi^{-6} hat{A}^{ij}) – frac{1}{3} mathcal{D}^i (mathcal{D}_j beta^j)$.

Además, en CFC, tenemos:

$latex hat{A}^{ij} = (LX)^{ij} + hat{A}^{ij}_{TT} approx (LX)^{ij} = mathcal{D}^i X^j + mathcal{D}^j X^i – frac{2}{3} mathcal{D}_k X^k f^{ij}$

donde $latex L$ es el operador de Killing conforme actuando sobre la parte longitudinal $latex X^i$ sin traza y $latex A^{ij}_{TT}$ es la parte transversal sin traza de la curvatura extrínseca , y de FCF tenemos:

  • la métrica inducida en cada hipersuperficie $latex gamma_{mu nu} := g_{mu nu} + n_{mu} n_{nu}$ (o $latex boldsymbol{gamma} := boldsymbol{g} + boldsymbol{n} otimes boldsymbol{n}$ ) con $latex boldsymbol{n} = frac{dt}{|dt|}$.
  • la curvatura extrínseca $latex boldsymbol{K:=-frac{1}{2}mathcal{L}_{boldsymbol{n}} boldsymbol{gamma}}$ (o, con índices, $latex K_{mu nu} = -frac{1}{2} mathcal{L}_{boldsymbol{n}} gamma_{mu nu}$).

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Como ya comentamos, de la tesis de Bauswein, adoptando la foliación $latex 3+1$ del espacio-tiempo la métrica queda:

$latex ds^2 = (- alpha^2 + beta_i beta^i) dt^2 + 2 beta_i dx^i dt + gamma_{ij} dx^i dx^j$

En la aproximación CFC resolvemos repetidamente el problema de valor inicial. De acuerdo con esta aproximación, la parte espacial de la métrica se puede escribir como:

$latex gamma_{ij} = psi^4 delta_{ij}$

donde $latex psi$ es el factor conforme (una transformación conforme preserva los ángulos. En geometría Riemanniana, dos métricas de Riemann $latex g$ y $latex h$ sobre una variedad $latex M$ son conformemente equivalentes si $latex g=uh$ para alguna función positiva $latex u$ sobre $latex M$. La función $latex u$ es el factor conforme).

De esta manera, las ecuaciones de Einstein, asumiendo $latex K := tr(K_{ij}) = K_i^i =0$, se reducen al sistema de cinco PDE elipticas no lineales acopladas:

$latex Delta psi = -2 pi psi^5 E – frac{1}{8} psi^5 K_{ij}K^{ij}$

$latex Delta(alpha psi) = 2 pi alpha psi^5 (E + 2S) + frac{7}{8} alpha psi^5 K_{ij}K^{ij}$

$latex Delta beta^i + frac{1}{3}partial^i partial_j beta^j = 16 pi alpha rho W + 2 psi^{10} K^{ij} partial_j (frac{alpha}{psi^6}) =: S_beta$

donde $latex E = rho h W^2 – P$, $latex S = rho h (W^2 -1) + 3P$ y

$latex K_{ij} = frac{psi^4}{2 alpha} (delta_{il} partial_j beta_l + delta_{jl} partial_i beta^l – frac{2}{3} delta_{ij} partial_k beta^k )$

que podemos escribir de manera mas compacta como:

$latex Delta B^i = S_beta$

$latex Delta chi = partial_i B^i$

si definimos $latex beta^i = B^i – frac{1}{4} partial_i chi$ y que es un sistema tipo Poisson que puede ser resuelto iterativamente hasta la convergencia con un método multigrid.

Las condiciones en la frontera se dan mediante desarrollo multipolar () de los terminos fuente, que son no compactas, hasta el armónico quadrupolar.

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