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Desacoplamiento de los sistemas para las X^i y \beta^i de la discretización en cartesianas de la reformulación covariante del sector elíptico de la aproximación CFC en términos de CoCoNuT

julio 1, 2013 in Física, Fluidos, Geometría, Matemáticas, PDE, Relatividad | Tags: , , , , | Deja un comentario

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

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

\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 \mathcal{D} conmuta con \Delta (métrica plana), tenemos:

\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:

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

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

\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:

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

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

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

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

donde inicialmente:

(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},

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

\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:

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

\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} =

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

¡que vuelve a ser lineal!

Continuamos con:

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

\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} =

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

y, finalmente:

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

\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} =

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

donde calculamos al principio:

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

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

(\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:

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

\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},

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

\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},

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

\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},

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

\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},

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

\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},

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

\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:

\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:

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

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

- \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:

\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} -

- \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:

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

y la ecuación:

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

+ \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:

\approx \frac{(\alpha\psi)_{i-1,j,k} - 2(\alpha\psi)_{i,j,k}+(\alpha\psi)_{i+1,j,k}}{h_x^2} +

+ \frac{(\alpha\psi)_{i,j-1,k}-2(\alpha\psi)_{i,j,k}+(\alpha\psi)_{i,j+1,k}}{h_y^2} +

+ \frac{(\alpha\psi)_{i,j,k-1}-2(\alpha\psi)_{i,j,k}+(\alpha\psi)_{i,j,k+1}}{h_z^2} =

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

+ \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:

\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}) +

- \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:

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:

\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 X^i:

\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:

\Delta \Theta_\beta = \frac{3}{4} \mathcal{D}^i (\mathcal{D}_j (2 \alpha \psi^{-6} \hat{A}^{ij})) =

\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:

\Theta_\beta := \mathcal{D}_i \beta^i,

que discretizada queda:

\frac{(\Theta_\beta)_{i-1,j,k}-2(\Theta_\beta)_{i,j,k}+(\Theta_\beta)_{i+1,j,k}}{h_x^2} +

\frac{(\Theta_\beta)_{i,j-1,k}-2(\Theta_\beta)_{i,j,k}+(\Theta_\beta)_{i,j+1,k}}{h_y^2} +

\frac{(\Theta_\beta)_{i,j,k-1}-2(\Theta_\beta)_{i,j,k}+(\Theta_\beta)_{i,j,k+1}}{h_z^2} =

\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:

\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}{3} \partial_x \Theta_\beta \approx

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

= (\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} -

- \frac{1}{3} (\partial_x \Theta_\beta)_{i,j,k}.

De la misma manera:

\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

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

= (\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} -

- \frac{1}{3} (\partial_y \Theta_\beta)_{i,j,k}.

Y, por último:

\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

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

= (\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} -

- \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…

Discretización de la reformulación covariante del sector elíptico de la aproximación CFC en términos de CoCoNuT

junio 6, 2013 in Física, Fluidos, Geometría, Matemáticas, PDE, Relatividad | Tags: , , , | 4 comentarios

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

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

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

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

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

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

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

\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},

\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},

\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:

\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

\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} =

= 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} +

+ \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} +

+ \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 F(X^x_{i,j,k})=0 y entonces tenemos:

\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}).

\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

\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} =

= 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} +

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

+ \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:

\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}).

\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

\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} =

= 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} +

+ \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} )

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

con:

\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:

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

\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},

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

\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},

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

\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},

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

\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},

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

\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},

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

\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:

\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:

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

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

- \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:

\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}) -

- \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:

\Delta (\alpha\psi) = 2 \pi (\alpha\psi)^{-1} ( D + \tau + 2 \rho h (w^2-1) + 6 p) +

+ \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:

\approx \frac{(\alpha\psi)_{i-1,j,k} - 2(\alpha\psi)_{i,j,k}+(\alpha\psi)_{i+1,j,k}}{h_x^2} + \frac{(\alpha\psi)_{i,j-1,k}-2(\alpha\psi)_{i,j,k}+(\alpha\psi)_{i,j+1,k}}{h_y^2} + \frac{(\alpha\psi)_{i,j,k-1}-2(\alpha\psi)_{i,j,k}+(\alpha\psi)_{i,j,k+1}}{h_z^2} =

=2 \pi (\alpha\psi)_{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}) +

+ \frac{7}{8}(\alpha\psi)_{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:

\partial_{\psi\alpha_{i,j,k}} F(\psi\alpha_{i,j,k}) = -2 ( \frac{1}{h_x^2} + \frac{1}{h_y^2} + \frac{1}{h_z^2} ) +

+ 2 \pi (\psi\alpha)_{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}) -

+ \frac{49}{8} (\psi\alpha)_{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:

\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}{3} \partial_x (\partial_x \beta^x + \partial_y \beta^y + \partial_z \beta^z) \approx

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

= \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} +

+ \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} +

+ \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} -

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

+ \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} +

+ \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:

\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}),

\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 (\partial_x \beta^x + \partial_y \beta^y + \partial_z \beta^z) \approx

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

= \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} +

+ \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} +

+ \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} -

- \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} +

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

+ \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:

\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}),

\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 (\partial_x \beta^x + \partial_y \beta^y + \partial_z \beta^z) \approx

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

= \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} +

+ \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} +

+ \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} -

- \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} +

+ \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} )

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

con:

\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} ).

Presentweet :-)

abril 22, 2013 in Física, Informática, Matemáticas, PDE, Relatividad, Supercomputación | Tags: , , , , | Deja un comentario

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Reescritura en cartesianas de la reformulación covariante del sector elíptico de la aproximación CFC en términos de CoCoNuT

marzo 25, 2013 in Física, Fluidos, Geometría, Matemáticas, PDE, Relatividad | Tags: , , , | 1 comentario

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

En primer lugar, teniamos:

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

donde:

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

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:

\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),

\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),

\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:

\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:

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

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

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

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

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

\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:

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

donde:

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

E:= D + \tau

es:

\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:

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

con:

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

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

queda:

\Delta (\alpha\psi) = 2 \pi (\alpha\psi)^{-1} ( D + \tau + 2 \rho h (w^2-1) + 6 p) +

+ \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)

Y la última:

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

que escribimos como:

\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}{3} \partial_x (\partial_x \beta^x + \partial_y \beta^y + \partial_z \beta^z)

\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 (\partial_x \beta^x + \partial_y \beta^y + \partial_z \beta^z)

\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 (\partial_x \beta^x + \partial_y \beta^y + \partial_z \beta^z)

Reescritura de la reformulación del sector elíptico de la aproximación CFC en términos de CoCoNuT

febrero 28, 2013 in Física, Fluidos, Geometría, Matemáticas, PDE, Relatividad | Tags: , , , | 1 comentario

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 h^{ij} sean cero, en terminos de las variables que éste utiliza. Empezamos con una auxilar:

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

donde:

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

S_j := \rho h w^2 v_j.

La primera es:

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

donde:

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

E:= D + \tau

La siguiente:

\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:

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

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

Y la última:

\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:

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

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

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