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} ).

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