1 Erratum to: CEAS Aeronaut J DOI 10.1007/s13272-012-0061-9

The original version of this article unfortunately contained mistakes. Equations 11, 12, and 13 were incorrect. The corrected equations are given below.

$$ \begin{aligned} {U^{i,j,k,t + 1}} &= - {U^{i,j,k,t - 1}} + 2{U^{i,j,k}}\\ & \quad- \frac{{2\chi }}{8}{U^{i,j,k}}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {( {\eta _x^2{{\tilde S}_{11}} + \eta _y^2{{\tilde S}_{66}} + \eta _z^2{{\tilde S}_{55}}} )}]}\\ & \quad+ \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {2\eta _x^2{{\tilde S}_{11}}{U^{i + \alpha ,j,k}} + 2\eta _y^2{{\tilde S}_{66}}{U^{i,j + \beta ,k}} + 2\eta _z^2{{\tilde S}_{55}}{U^{i,j,k + \gamma }}} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \beta {\eta _x}{\eta _y}( {{{\tilde S}_{12}} + {{\tilde S}_{66}}})( {{V^{i + \alpha ,j + \beta ,k}} - {V^{i,j,k}}} )} ]}\\ & \quad+ \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \beta {\eta _x}{\eta _y}( {{{\tilde S}_{12}} - {{\tilde S}_{66}}} )( {{V^{i,j + \beta ,k}} - {V^{i + \alpha ,j,k}}} )} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}( {{{\tilde S}_{13}} + {{\tilde S}_{55}}} )( {{W^{i + \alpha ,j,k + \gamma }} - {W^{i,j,k}}} )} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}( {{{\tilde S}_{13}} - {{\tilde S}_{55}}} )( {{W^{i,j,k + \gamma }} - {W^{i + \alpha ,j,k}}} )}]}\\ & \quad- \frac{{2\chi }}{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \beta {\eta _x}{\eta _y}{{\tilde S}_{16}}( {{U^{i,j,k}} - {U^{i + \alpha ,j + \beta ,k}}})}]}\\ &\quad - \frac{{2\chi }}{8}{V^{i,j,k}}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\eta _x^2{{\tilde S}_{16}} + \eta _y^2{{\tilde S}_{26}}} ]}\\ &\quad - \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}( {{{\tilde S}_{36}} + {{\tilde S}_{45}}} ){W^{i,j,k}}} ]}\\ &\quad + \frac{{2\chi }}{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\eta _x^2{{\tilde S}_{16}}{V^{i + \alpha ,j,k}} + \eta _y^2{{\tilde S}_{26}}{V^{i,j + \beta ,k}}} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}{{\tilde S}_{36}}( {{W^{i,j + \beta ,k + \gamma }} + {W^{i,j,k + \gamma }} - {W^{i,j + \beta ,k}}})} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}{{\tilde S}_{45}}( {{W^{i,j + \beta ,k + \gamma }} - {W^{i,j,k + \gamma }} + {W^{i,j + \beta ,k}}} )}]}\\ & \quad+ \frac{{2\chi }}{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {\eta _z^2{{\tilde S}_{45}}( {{V^{i,j,k + \gamma }} - {V^{i,j,k}}})} \end{aligned} $$
(11)
$$ \begin{aligned} {V^{i,j,k,t + 1}} &= - {V^{i,j,k,t - 1}} + 2{V^{i,j,k}}\\ &\quad - \frac{{2\chi }}{8}{V^{i,j,k}}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\eta _x^2{{\tilde S}_{66}} + \eta _y^2{{\tilde S}_{22}} + \eta _z^2{{\tilde S}_{44}}} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {2\eta _x^2{{\tilde S}_{66}}{V^{i + \alpha ,j,k}} + 2\eta _y^2{{\tilde S}_{22}}{V^{i,j + \beta ,k}} + 2\eta _z^2{{\tilde S}_{44}}{V^{i,j,k + \gamma }}}]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \beta {\eta _x}{\eta _y}( {{{\tilde S}_{12}} + {{\tilde S}_{66}}} )( {{U^{i + \alpha ,j + \beta ,k}} - {U^{i,j,k}}} )} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \beta {\eta _x}{\eta _y}( {{{\tilde S}_{12}} - {{\tilde S}_{66}}} )( {{U^{i + \alpha ,j,k}} - {U^{i,j + \beta ,k}}} )} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}( {{{\tilde S}_{23}} + {{\tilde S}_{44}}} )( {{W^{i,j + \beta ,k + \gamma }} - {W^{i,j,k}}} )} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}( {{{\tilde S}_{23}} - {{\tilde S}_{44}}} )( {{W^{i,j,k + \gamma }} - {W^{i,j + \beta ,k}}} )}]}\\ &\quad - \frac{{2\chi }}{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \beta {\eta _x}{\eta _y}{{\tilde S}_{26}}( {{V^{i,j,k}} - {V^{i + \alpha ,j + \beta ,k}}} )} ]}\\ &\quad - \frac{{2\chi }}{8}{U^{i,j,k}}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\eta _x^2{{\tilde S}_{16}} + \eta _y^2{{\tilde S}_{26}}} ]}\\ & \quad+ \frac{{2\chi }}{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\eta _x^2{{\tilde S}_{16}}{U^{i + \alpha ,j,k}} + \eta _y^2{{\tilde S}_{26}}{U^{i,j + \beta ,k}}} ]}\\ &\quad - \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}( {{{\tilde S}_{36}} + {{\tilde S}_{45}}} ){W^{i,j,k}}} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}{{\tilde S}_{36}}( {{W^{i+ \alpha,j ,k + \gamma }} + {W^{i,j,k + \gamma }} - {W^{i + \alpha ,j,k}}} )} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}{{\tilde S}_{45}}( {{W^{i+ \alpha,j ,k + \gamma }} - {W^{i,j,k + \gamma }} + {W^{i + \alpha ,j,k}}} )} ]}\\ &\quad + \frac{{2\chi }}{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\eta _z^2{{\tilde S}_{45}}( {{U^{i,j,k + \gamma }} - {U^{i,j,k}}} )} ]} \end{aligned} $$
(12)
$$ \begin{aligned} {W^{i,j,k,t + 1}} &= - {W^{i,j,k,t - 1}} + 2{W^{i,j,k}}\\ &\quad - \frac{{2\chi }}{8}{W^{i,j,k}}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\eta _x^2{{\tilde S}_{55}} + \eta _y^2{{\tilde S}_{44}} + \eta _z^2{{\tilde S}_{33}}} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {2\eta _x^2{{\tilde S}_{55}}{W^{i + \alpha ,j,k}} + 2\eta _y^2{{\tilde S}_{44}}{W^{i,j + \beta ,k}} + 2\eta _z^2{{\tilde S}_{33}}{W^{i,j,k + \gamma }}} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}( {{{\tilde S}_{23}} + {{\tilde S}_{44}}} )( {{V^{i,j + \beta ,k + \gamma }} - {V^{i,j,k}}})} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}( {{{\tilde S}_{23}} - {{\tilde S}_{44}}} )( {{V^{i,j + \beta ,k}} - {V^{i,j,k + \gamma }}} )} ]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}( {{{\tilde S}_{13}} + {{\tilde S}_{55}}} )( {{U^{i + \alpha ,j,k + \gamma }} - {U^{i,j,k}}} )}]}\\ &\quad + \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}( {{{\tilde S}_{13}} - {{\tilde S}_{55}}} )( {{U^{i + \alpha ,j,k}} - {U^{i,j,k + \gamma }}} )} ]}\\ &\quad - \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}( {{{\tilde S}_{36}} + {{\tilde S}_{45}}} )( {{U^{i,j,k}} - {U^{i,j + \beta ,k + \gamma }}} )} ]}\\ & \quad- \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}( {{{\tilde S}_{36}} + {{\tilde S}_{45}}} )( {{V^{i,j,k}} - {V^{i + \alpha ,j,k + \gamma }}} )}]}\\ &\quad - \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\beta \gamma {\eta _y}{\eta _z}( {{{\tilde S}_{36}} - {{\tilde S}_{45}}} )( {{U^{i,j,k + \gamma }} - {U^{i,j + \beta ,k}}} )} ]}\\ &\quad - \frac{\chi }{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \gamma {\eta _x}{\eta _z}( {{{\tilde S}_{36}} - {{\tilde S}_{45}}} )( {{V^{i,j,k + \gamma }} - {V^{i + \alpha ,j,k}}} )} ]}\\ &\quad + \frac{{2\chi }}{8}\sum\limits_{\alpha ,\beta ,\gamma = \pm 1}^{} {[ {\alpha \beta {\eta _x}{\eta _y}{{\tilde S}_{45}}( {{W^{i + \alpha ,j + \beta ,k}} - {W^{i,j,k}}} )} ]} \end{aligned} $$
(13)