有限要素法・流体力学による数値計算

\[ \phi= \frac{\partial P}{\partial\tau} +V_x\frac{\partial P}{\partial x} +V_y\frac{\partial P}{\partial y} +V_z\frac{\partial P}{\partial z} +\frac{1}{Ma^2}\left( \frac{\partial V_x}{\partial X} +\frac{\partial V_y}{\partial Y} +\frac{\partial V_z}{\partial Z} \right)=0 \]

\[ \begin{aligned} &\int_V[N]^T\phi\,dV =\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix}\phi\,dV\\[6pt] &={}\int_V \begin{bmatrix}0\\0\\0\\0\end{bmatrix}dV\\[6pt] &={}\begin{bmatrix}0\\0\\0\\0\end{bmatrix} \end{aligned} \]

\[ \begin{aligned} V_x &=N_1V_{x1}+N_2V_{x2}+N_3V_{x3}+N_4V_{x4}\\[6pt] &=\begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{Bmatrix}V_{x1}&V_{x2}&V_{x3}&V_{x4}\end{Bmatrix}\\[6pt] &=[N]^T\{V_x\}\\[6pt] V_y&=[N]^T\{V_y\}\\[6pt] V_z&=[N]^T\{V_z\}\\[6pt] P&=[N]^T\{P\} \end{aligned} \]

\[ \begin{aligned} &\int_V[N]^T\phi\,dV\\[6pt] &={}\int_V[N]^T\left\{ \frac{\partial P}{\partial\tau} +V_x\frac{\partial P}{\partial x} +V_y\frac{\partial P}{\partial y} +V_z\frac{\partial P}{\partial z} +\frac{1}{Ma^2}\left( \frac{\partial V_x}{\partial X} +\frac{\partial V_y}{\partial Y} +\frac{\partial V_z}{\partial Z} \right) \right\}dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T\frac{\partial P}{\partial\tau}\,dV +\int_V[N]^TV_x\frac{\partial P}{\partial X}\,dV +\int_V[N]^TV_y\frac{\partial P}{\partial Y}\,dV +\int_V[N]^TV_z\frac{\partial P}{\partial Z}\,dV\\[6pt] &+\frac{1}{Ma^2}\int_V[N]^T\frac{\partial V_x}{\partial X}\,dV +\frac{1}{Ma^2}\int_V[N]^T\frac{\partial V_y}{\partial Y}\,dV +\frac{1}{Ma^2}\int_V[N]^T\frac{\partial V_z}{\partial Z}\,dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\frac{\partial[N]\{P\}^{\Delta\tau+\tau}}{\partial X}dV +V_y\int_V[N]^T\frac{\partial[N]\{P\}^{\Delta\tau+\tau}}{\partial Y}dV\\[6pt] &+V_z\int_V[N]^T\frac{\partial[N]\{P\}^{\Delta\tau+\tau}}{\partial Z}dV\\[6pt] &+\frac{1}{Ma^2}\int_V[N]^T\frac{\partial[N]\{V_x\}^{\Delta\tau+\tau}}{\partial X}dV +\frac{1}{Ma^2}\int_V[N]^T\frac{\partial[N]\{V_y\}^{\Delta\tau+\tau}}{\partial Y}dV\\[6pt] &+\frac{1}{Ma^2}\int_V[N]^T\frac{\partial[N]\{V_z\}^{\Delta\tau+\tau}}{\partial Z}dV \end{aligned} \]

\[ \begin{aligned} &={}\int_V[N]^T[N]dV\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V[N]^T\frac{\partial[N]}{\partial X}dV\,\{P\}^{\Delta\tau+\tau} +V_y\int_V[N]^T\frac{\partial[N]}{\partial Y}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_z\int_V[N]^T\frac{\partial[N]}{\partial Z}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_V[N]^T\frac{\partial[N]}{\partial X}dV\,\{V_x\}^{\Delta\tau+\tau} +\frac{1}{Ma^2}\int_V[N]^T\frac{\partial[N]}{\partial Y}dV\,\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+\frac{1}{Ma^2}\int_V[N]^T\frac{\partial[N]}{\partial Z}dV\,\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{bmatrix}N_1&N_2&N_3&N_4\end{bmatrix}dV\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial X}& \dfrac{\partial N_2}{\partial X}& \dfrac{\partial N_3}{\partial X}& \dfrac{\partial N_4}{\partial X} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial Y}& \dfrac{\partial N_2}{\partial Y}& \dfrac{\partial N_3}{\partial Y}& \dfrac{\partial N_4}{\partial Y} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_z\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial Z}& \dfrac{\partial N_2}{\partial Z}& \dfrac{\partial N_3}{\partial Z}& \dfrac{\partial N_4}{\partial Z} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial X}& \dfrac{\partial N_2}{\partial X}& \dfrac{\partial N_3}{\partial X}& \dfrac{\partial N_4}{\partial X} \end{bmatrix}dV\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial Y}& \dfrac{\partial N_2}{\partial Y}& \dfrac{\partial N_3}{\partial Y}& \dfrac{\partial N_4}{\partial Y} \end{bmatrix}dV\,\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix}N_1\\N_2\\N_3\\N_4\end{bmatrix} \begin{bmatrix} \dfrac{\partial N_1}{\partial Z}& \dfrac{\partial N_2}{\partial Z}& \dfrac{\partial N_3}{\partial Z}& \dfrac{\partial N_4}{\partial Z} \end{bmatrix}dV\,\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_V \begin{bmatrix} N_1N_1&N_1N_2&N_1N_3&N_1N_4\\ N_2N_1&N_2N_2&N_2N_3&N_2N_4\\ N_3N_1&N_3N_2&N_3N_3&N_3N_4\\ N_4N_1&N_4N_2&N_4N_3&N_4N_4 \end{bmatrix}dV\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial X}&N_1\dfrac{\partial N_2}{\partial X}&N_1\dfrac{\partial N_3}{\partial X}&N_1\dfrac{\partial N_4}{\partial X}\\ N_2\dfrac{\partial N_1}{\partial X}&N_2\dfrac{\partial N_2}{\partial X}&N_2\dfrac{\partial N_3}{\partial X}&N_2\dfrac{\partial N_4}{\partial X}\\ N_3\dfrac{\partial N_1}{\partial X}&N_3\dfrac{\partial N_2}{\partial X}&N_3\dfrac{\partial N_3}{\partial X}&N_3\dfrac{\partial N_4}{\partial X}\\ N_4\dfrac{\partial N_1}{\partial X}&N_4\dfrac{\partial N_2}{\partial X}&N_4\dfrac{\partial N_3}{\partial X}&N_4\dfrac{\partial N_4}{\partial X} \end{bmatrix}dV\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_V \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial Y}&N_1\dfrac{\partial N_2}{\partial Y}&N_1\dfrac{\partial N_3}{\partial Y}&N_1\dfrac{\partial N_4}{\partial Y}\\ N_2\dfrac{\partial N_1}{\partial Y}&N_2\dfrac{\partial N_2}{\partial Y}&N_2\dfrac{\partial N_3}{\partial Y}&N_2\dfrac{\partial N_4}{\partial Y}\\ N_3\dfrac{\partial N_1}{\partial Y}&N_3\dfrac{\partial N_2}{\partial Y}&N_3\dfrac{\partial N_3}{\partial Y}&N_3\dfrac{\partial N_4}{\partial Y}\\ N_4\dfrac{\partial N_1}{\partial Y}&N_4\dfrac{\partial N_2}{\partial Y}&N_4\dfrac{\partial N_3}{\partial Y}&N_4\dfrac{\partial N_4}{\partial Y} \end{bmatrix}dV\,\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+V_z\int_V \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial Z}&N_1\dfrac{\partial N_2}{\partial Z}&N_1\dfrac{\partial N_3}{\partial Z}&N_1\dfrac{\partial N_4}{\partial Z}\\ N_2\dfrac{\partial N_1}{\partial Z}&N_2\dfrac{\partial N_2}{\partial Z}&N_2\dfrac{\partial N_3}{\partial Z}&N_2\dfrac{\partial N_4}{\partial Z}\\ N_3\dfrac{\partial N_1}{\partial Z}&N_3\dfrac{\partial N_2}{\partial Z}&N_3\dfrac{\partial N_3}{\partial Z}&N_3\dfrac{\partial N_4}{\partial Z}\\ N_4\dfrac{\partial N_1}{\partial Z}&N_4\dfrac{\partial N_2}{\partial Z}&N_4\dfrac{\partial N_3}{\partial Z}&N_4\dfrac{\partial N_4}{\partial Z} \end{bmatrix}dV\,\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &+\frac1{Ma^2}\int_V \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial X}&N_1\dfrac{\partial N_2}{\partial X}&N_1\dfrac{\partial N_3}{\partial X}&N_1\dfrac{\partial N_4}{\partial X}\\ N_2\dfrac{\partial N_1}{\partial X}&N_2\dfrac{\partial N_2}{\partial X}&N_2\dfrac{\partial N_3}{\partial X}&N_2\dfrac{\partial N_4}{\partial X}\\ N_3\dfrac{\partial N_1}{\partial X}&N_3\dfrac{\partial N_2}{\partial X}&N_3\dfrac{\partial N_3}{\partial X}&N_3\dfrac{\partial N_4}{\partial X}\\ N_4\dfrac{\partial N_1}{\partial X}&N_4\dfrac{\partial N_2}{\partial X}&N_4\dfrac{\partial N_3}{\partial X}&N_4\dfrac{\partial N_4}{\partial X} \end{bmatrix}dV\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial Y}&N_1\dfrac{\partial N_2}{\partial Y}&N_1\dfrac{\partial N_3}{\partial Y}&N_1\dfrac{\partial N_4}{\partial Y}\\ N_2\dfrac{\partial N_1}{\partial Y}&N_2\dfrac{\partial N_2}{\partial Y}&N_2\dfrac{\partial N_3}{\partial Y}&N_2\dfrac{\partial N_4}{\partial Y}\\ N_3\dfrac{\partial N_1}{\partial Y}&N_3\dfrac{\partial N_2}{\partial Y}&N_3\dfrac{\partial N_3}{\partial Y}&N_3\dfrac{\partial N_4}{\partial Y}\\ N_4\dfrac{\partial N_1}{\partial Y}&N_4\dfrac{\partial N_2}{\partial Y}&N_4\dfrac{\partial N_3}{\partial Y}&N_4\dfrac{\partial N_4}{\partial Y} \end{bmatrix}dV\,\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix} N_1\dfrac{\partial N_1}{\partial Z}&N_1\dfrac{\partial N_2}{\partial Z}&N_1\dfrac{\partial N_3}{\partial Z}&N_1\dfrac{\partial N_4}{\partial Z}\\ N_2\dfrac{\partial N_1}{\partial Z}&N_2\dfrac{\partial N_2}{\partial Z}&N_2\dfrac{\partial N_3}{\partial Z}&N_2\dfrac{\partial N_4}{\partial Z}\\ N_3\dfrac{\partial N_1}{\partial Z}&N_3\dfrac{\partial N_2}{\partial Z}&N_3\dfrac{\partial N_3}{\partial Z}&N_3\dfrac{\partial N_4}{\partial Z}\\ N_4\dfrac{\partial N_1}{\partial Z}&N_4\dfrac{\partial N_2}{\partial Z}&N_4\dfrac{\partial N_3}{\partial Z}&N_4\dfrac{\partial N_4}{\partial Z} \end{bmatrix}dV\,\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_V \begin{bmatrix} L_1L_1&L_1L_2&L_1L_3&L_1L_4\\ L_2L_1&L_2L_2&L_2L_3&L_2L_4\\ L_3L_1&L_3L_2&L_3L_3&L_3L_4\\ L_4L_1&L_4L_2&L_4L_3&L_4L_4 \end{bmatrix}dV\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial X}&L_1\dfrac{\partial L_2}{\partial X}&L_1\dfrac{\partial L_3}{\partial X}&L_1\dfrac{\partial L_4}{\partial X}\\ L_2\dfrac{\partial L_1}{\partial X}&L_2\dfrac{\partial L_2}{\partial X}&L_2\dfrac{\partial L_3}{\partial X}&L_2\dfrac{\partial L_4}{\partial X}\\ L_3\dfrac{\partial L_1}{\partial X}&L_3\dfrac{\partial L_2}{\partial X}&L_3\dfrac{\partial L_3}{\partial X}&L_3\dfrac{\partial L_4}{\partial X}\\ L_4\dfrac{\partial L_1}{\partial X}&L_4\dfrac{\partial L_2}{\partial X}&L_4\dfrac{\partial L_3}{\partial X}&L_4\dfrac{\partial L_4}{\partial X} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_V \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial Y}&L_1\dfrac{\partial L_2}{\partial Y}&L_1\dfrac{\partial L_3}{\partial Y}&L_1\dfrac{\partial L_4}{\partial Y}\\ L_2\dfrac{\partial L_1}{\partial Y}&L_2\dfrac{\partial L_2}{\partial Y}&L_2\dfrac{\partial L_3}{\partial Y}&L_2\dfrac{\partial L_4}{\partial Y}\\ L_3\dfrac{\partial L_1}{\partial Y}&L_3\dfrac{\partial L_2}{\partial Y}&L_3\dfrac{\partial L_3}{\partial Y}&L_3\dfrac{\partial L_4}{\partial Y}\\ L_4\dfrac{\partial L_1}{\partial Y}&L_4\dfrac{\partial L_2}{\partial Y}&L_4\dfrac{\partial L_3}{\partial Y}&L_4\dfrac{\partial L_4}{\partial Y} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_z\int_V \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial Z}&L_1\dfrac{\partial L_2}{\partial Z}&L_1\dfrac{\partial L_3}{\partial Z}&L_1\dfrac{\partial L_4}{\partial Z}\\ L_2\dfrac{\partial L_1}{\partial Z}&L_2\dfrac{\partial L_2}{\partial Z}&L_2\dfrac{\partial L_3}{\partial Z}&L_2\dfrac{\partial L_4}{\partial Z}\\ L_3\dfrac{\partial L_1}{\partial Z}&L_3\dfrac{\partial L_2}{\partial Z}&L_3\dfrac{\partial L_3}{\partial Z}&L_3\dfrac{\partial L_4}{\partial Z}\\ L_4\dfrac{\partial L_1}{\partial Z}&L_4\dfrac{\partial L_2}{\partial Z}&L_4\dfrac{\partial L_3}{\partial Z}&L_4\dfrac{\partial L_4}{\partial Z} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial X}&L_1\dfrac{\partial L_2}{\partial X}&L_1\dfrac{\partial L_3}{\partial X}&L_1\dfrac{\partial L_4}{\partial X}\\ L_2\dfrac{\partial L_1}{\partial X}&L_2\dfrac{\partial L_2}{\partial X}&L_2\dfrac{\partial L_3}{\partial X}&L_2\dfrac{\partial L_4}{\partial X}\\ L_3\dfrac{\partial L_1}{\partial X}&L_3\dfrac{\partial L_2}{\partial X}&L_3\dfrac{\partial L_3}{\partial X}&L_3\dfrac{\partial L_4}{\partial X}\\ L_4\dfrac{\partial L_1}{\partial X}&L_4\dfrac{\partial L_2}{\partial X}&L_4\dfrac{\partial L_3}{\partial X}&L_4\dfrac{\partial L_4}{\partial X} \end{bmatrix}dV\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial Y}&L_1\dfrac{\partial L_2}{\partial Y}&L_1\dfrac{\partial L_3}{\partial Y}&L_1\dfrac{\partial L_4}{\partial Y}\\ L_2\dfrac{\partial L_1}{\partial Y}&L_2\dfrac{\partial L_2}{\partial Y}&L_2\dfrac{\partial L_3}{\partial Y}&L_2\dfrac{\partial L_4}{\partial Y}\\ L_3\dfrac{\partial L_1}{\partial Y}&L_3\dfrac{\partial L_2}{\partial Y}&L_3\dfrac{\partial L_3}{\partial Y}&L_3\dfrac{\partial L_4}{\partial Y}\\ L_4\dfrac{\partial L_1}{\partial Y}&L_4\dfrac{\partial L_2}{\partial Y}&L_4\dfrac{\partial L_3}{\partial Y}&L_4\dfrac{\partial L_4}{\partial Y} \end{bmatrix}dV\,\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V \begin{bmatrix} L_1\dfrac{\partial L_1}{\partial Z}&L_1\dfrac{\partial L_2}{\partial Z}&L_1\dfrac{\partial L_3}{\partial Z}&L_1\dfrac{\partial L_4}{\partial Z}\\ L_2\dfrac{\partial L_1}{\partial Z}&L_2\dfrac{\partial L_2}{\partial Z}&L_2\dfrac{\partial L_3}{\partial Z}&L_2\dfrac{\partial L_4}{\partial Z}\\ L_3\dfrac{\partial L_1}{\partial Z}&L_3\dfrac{\partial L_2}{\partial Z}&L_3\dfrac{\partial L_3}{\partial Z}&L_3\dfrac{\partial L_4}{\partial Z}\\ L_4\dfrac{\partial L_1}{\partial Z}&L_4\dfrac{\partial L_2}{\partial Z}&L_4\dfrac{\partial L_3}{\partial Z}&L_4\dfrac{\partial L_4}{\partial Z} \end{bmatrix}dV\,\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\int_V \begin{bmatrix} L_1L_1&L_1L_2&L_1L_3&L_1L_4\\ L_2L_1&L_2L_2&L_2L_3&L_2L_4\\ L_3L_1&L_3L_2&L_3L_3&L_3L_4\\ L_4L_1&L_4L_2&L_4L_3&L_4L_4 \end{bmatrix}dV\, \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x\int_V\frac1{6V} \begin{bmatrix} L_1c_{1x}&L_1c_{2x}&L_1c_{3x}&L_1c_{4x}\\ L_2c_{1x}&L_2c_{2x}&L_2c_{3x}&L_2c_{4x}\\ L_3c_{1x}&L_3c_{2x}&L_3c_{3x}&L_3c_{4x}\\ L_4c_{1x}&L_4c_{2x}&L_4c_{3x}&L_4c_{4x} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_y\int_V\frac1{6V} \begin{bmatrix} L_1c_{1y}&L_1c_{2y}&L_1c_{3y}&L_1c_{4y}\\ L_2c_{1y}&L_2c_{2y}&L_2c_{3y}&L_2c_{4y}\\ L_3c_{1y}&L_3c_{2y}&L_3c_{3y}&L_3c_{4y}\\ L_4c_{1y}&L_4c_{2y}&L_4c_{3y}&L_4c_{4y} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_z\int_V\frac1{6V} \begin{bmatrix} L_1c_{1z}&L_1c_{2z}&L_1c_{3z}&L_1c_{4z}\\ L_2c_{1z}&L_2c_{2z}&L_2c_{3z}&L_2c_{4z}\\ L_3c_{1z}&L_3c_{2z}&L_3c_{3z}&L_3c_{4z}\\ L_4c_{1z}&L_4c_{2z}&L_4c_{3z}&L_4c_{4z} \end{bmatrix}dV\,\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V\frac1{6V} \begin{bmatrix} L_1c_{1x}&L_1c_{2x}&L_1c_{3x}&L_1c_{4x}\\ L_2c_{1x}&L_2c_{2x}&L_2c_{3x}&L_2c_{4x}\\ L_3c_{1x}&L_3c_{2x}&L_3c_{3x}&L_3c_{4x}\\ L_4c_{1x}&L_4c_{2x}&L_4c_{3x}&L_4c_{4x} \end{bmatrix}dV\,\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V\frac1{6V} \begin{bmatrix} L_1c_{1y}&L_1c_{2y}&L_1c_{3y}&L_1c_{4y}\\ L_2c_{1y}&L_2c_{2y}&L_2c_{3y}&L_2c_{4y}\\ L_3c_{1y}&L_3c_{2y}&L_3c_{3y}&L_3c_{4y}\\ L_4c_{1y}&L_4c_{2y}&L_4c_{3y}&L_4c_{4y} \end{bmatrix}dV\,\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\int_V\frac1{6V} \begin{bmatrix} L_1c_{1z}&L_1c_{2z}&L_1c_{3z}&L_1c_{4z}\\ L_2c_{1z}&L_2c_{2z}&L_2c_{3z}&L_2c_{4z}\\ L_3c_{1z}&L_3c_{2z}&L_3c_{3z}&L_3c_{4z}\\ L_4c_{1z}&L_4c_{2z}&L_4c_{3z}&L_4c_{4z} \end{bmatrix}dV\,\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \int_VL_1^pL_2^qL_3^rL_4^s\,dV =\frac{p!q!r!s!}{(p+q+r+s+3)!}\,6V \] \[ \int_VL_iL_j\,dV= \begin{cases} \dfrac{1!1!}{(1+1+3)!}\,6V =\dfrac{6}{5!}V =\dfrac{1}{20}V & (i\ne j)\\[6pt] \dfrac{2!}{(1+1+3)!}\,6V =\dfrac{12}{5!}V =\dfrac{1}{10}V & (i=j) \end{cases} \] \[ \int_VL_i\,dV =\frac{1!}{(1+3)!}\,6V =\frac{6}{4!}V =\frac14V \]

\[ \begin{aligned} &={}\frac{V}{20} \begin{bmatrix} 2&1&1&1\\ 1&2&1&1\\ 1&1&2&1\\ 1&1&1&2 \end{bmatrix} \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x} \end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y} \end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z} \end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x} \end{bmatrix}\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y} \end{bmatrix}\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\frac1{6V}\frac{V}{4} \begin{bmatrix} c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z} \end{bmatrix}\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}\frac{V}{20} \begin{bmatrix} 2&1&1&1\\ 1&2&1&1\\ 1&1&2&1\\ 1&1&1&2 \end{bmatrix} \frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+\frac1{24} \begin{bmatrix} c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x} \end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{24} \begin{bmatrix} c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y} \end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{24} \begin{bmatrix} c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z} \end{bmatrix}\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\frac1{24} \begin{bmatrix} c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x}\\ c_{1x}&c_{2x}&c_{3x}&c_{4x} \end{bmatrix}\{V_x\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\frac1{24} \begin{bmatrix} c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y}\\ c_{1y}&c_{2y}&c_{3y}&c_{4y} \end{bmatrix}\{V_y\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\frac1{24} \begin{bmatrix} c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z}\\ c_{1z}&c_{2z}&c_{3z}&c_{4z} \end{bmatrix}\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ \begin{aligned} &={}[C]\frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau}\\[6pt] &+V_x[C_x]\{P\}^{\Delta\tau+\tau} +V_y[C_y]\{P\}^{\Delta\tau+\tau} +V_z[C_z]\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}[C_x]\{V_x\}^{\Delta\tau+\tau} +\frac1{Ma^2}[C_y]\{V_y\}^{\Delta\tau+\tau} +\frac1{Ma^2}[C_z]\{V_z\}^{\Delta\tau+\tau} \end{aligned} \]

\[ =\begin{bmatrix}0\\0\\0\\0\end{bmatrix} \]

\[ \begin{aligned} &[C]\frac{\{P\}^{\Delta\tau+\tau}-\{P\}^{\tau}}{\Delta\tau} +V_x[C_x]\{P\}^{\Delta\tau+\tau} +V_y[C_y]\{P\}^{\Delta\tau+\tau} +V_z[C_z]\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}[C_x]\{V_x\}^{\Delta\tau+\tau} +\frac1{Ma^2}[C_y]\{V_y\}^{\Delta\tau+\tau} +\frac1{Ma^2}[C_z]\{V_z\}^{\Delta\tau+\tau}=0 \end{aligned} \]

\[ \begin{aligned} &\frac{[C]}{\Delta\tau}\{P\}^{\Delta\tau+\tau}\\[6pt] &+V_x[C_x]\{P\}^{\Delta\tau+\tau} +V_y[C_y]\{P\}^{\Delta\tau+\tau} +V_z[C_z]\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}[C_x]\{V_x\}^{\Delta\tau+\tau} +\frac1{Ma^2}[C_y]\{V_y\}^{\Delta\tau+\tau} +\frac1{Ma^2}[C_z]\{V_z\}^{\Delta\tau+\tau} =\frac{[C]}{\Delta\tau}\{P\}^{\tau} \end{aligned} \]

\[ \begin{aligned} &\left( \frac{[C]}{\Delta\tau} +V_x[C_x]+V_y[C_y]+V_z[C_z] \right)\{P\}^{\Delta\tau+\tau}\\[6pt] &+\frac1{Ma^2}\left( [C_x]\{V_x\}^{\Delta\tau+\tau} +[C_y]\{V_y\}^{\Delta\tau+\tau} +[C_z]\{V_z\}^{\Delta\tau+\tau} \right) =\frac{[C]}{\Delta\tau}\{P\}^{\tau} \end{aligned} \]