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

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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