User:Arthurv~enwiki/Notebook

Diagonalized Flux Jacobian


A_x= \left[ \begin{array}{c c c c c} 0 & 1 & 0 & 0 & 0 \\ \hat{\gamma}H-u^2-a^2 & (3-\gamma)u & -v\hat{\gamma} & -w\hat{\gamma} & \hat{\gamma} \\ -uv & v & u & 0 & 0 \\ -uw & w & 0 & u & 0 \\ u[(\gamma-2)H-a^2] & H-\hat{\gamma}u^2 & -uv\hat{\gamma} & -uw\hat{\gamma} & \gamma u \end{array} \right] $$



P_x= \left[ \begin{array}{c c c c c} 1 &   1 & 0 & 0 & 1 \\ u-a & u & 0 & 0 & u+a \\ v &   v & 1 & 0 & v \\ w&   w & 0 &  1  & w \\ H-ua & \frac{1}{2}\vec{V}^2 & v & w & H+ua \end{array} \right] $$



\mathbf{\Lambda}_x= \left[ \begin{array}{c c c c c} u-a &  &   &   &   \\ & u &  &   &   \\ &  & u &   &   \\ &   &   & u  &   \\ &  &   &   & u+a \end{array} \right] $$


 * $$P^{-1}_x= \left[

\begin{array}{c c c c c} \dfrac{1}{2}\vec{V}^2+\dfrac{ua}{\gamma-1} & -u -\dfrac{a}{\gamma-1} & -v & -w & 1 \\ \\ 2\left(\dfrac{a^2}{\gamma-1}-\dfrac{\vec{V}^2}{2}\right) & 2u & 2v & 2w & -2 \\ \\ -v\dfrac{2a^2 }{\gamma-1} & 0 & \dfrac{2a^2}{\gamma-1} & 0 & 0 \\ \\ -w\dfrac{2a^2 }{\gamma-1} & 0 & 0 & \dfrac{2a^2}{\gamma-1} & 0 \\ \\ \dfrac{1}{2}\vec{V}^2-\dfrac{ua}{\gamma-1} & \dfrac{a}{\gamma-1}-u & -v & -w & 1 \end{array} \right]\frac{\gamma-1}{2a^2} $$


 * $$P_y= \left[

\begin{array}{c c c c c} 1 &   1 & 0 & 0 & 1 \\ u &   u & 1 & 0 & u \\ v-a & v & 0 & 0 & v+a \\ w&   w & 0 &  1  & w \\ H-va & \frac{1}{2}\vec{V}^2 & u & w & H+va \end{array} \right] $$

Flux Jacobian in arbitrary orientation:
$$\mathbf{A}_{IJ} = \frac{\partial E}{\partial x}n_x + \frac{\partial F}{\partial y}n_y + \frac{\partial G}{\partial z}n_z$$



A_{IJ}= \left[ \begin{array}{c c c c c} 0& n_x & n_y & n_z & 0 \\ \\ \left[\hat{\gamma}H-a^2\right]n_x-u\vec{V}  &   (2-\gamma)un_x+\vec{V} &    un_y-v\hat{\gamma}n_x        &   un_z-w\hat{\gamma}n_x  &\hat{\gamma}n_x\\ \\ \left[\hat{\gamma}H-a^2\right]n_y-v\vec{V}  &   vn_x-u\hat{\gamma}n_y        &    (2-\gamma)vn_y+\vec{V} &   vn_z-w\hat{\gamma}n_y & \hat{\gamma}n_y \\ \\ \left[\hat{\gamma}H-a^2\right]n_z-w\vec{V}  &   wn_x-u\hat{\gamma}n_z       &    wn_y-v\hat{\gamma}n_z        &   (2-\gamma)wn_z+\vec{V}& \hat{\gamma}n_z \\ \\ \vec{V}\left[(\gamma-2)H-a^2\right] & Hn_x-u\vec{V}\hat{\gamma} & Hn_y-v\vec{V}\hat{\gamma} & Hn_z-w\vec{V}\hat{\gamma} & \gamma\vec{V} \end{array} \right] $$

Diagonalizing:


 * $$\mathbf{A}_{IJ}=\mathbf{P\Lambda P}^{-1}$$


 * $$\mathbf{P}=(R_1,R_2,R_3,R_4,R_5)$$


 * $$R_1 = \left[

\begin{array}{c} 1 \\ u+an_x \\ v+an_y \\ w+an_z \\ H+\vec{V}a \vec{n} \end{array}\right] $$ corresponding to eigenvalue $$\lambda_1=|\vec{V}+a|$$


 * $$R_2 = \left[

\begin{array}{c} 1 \\ u-an_x \\ v-an_y \\ w-an_z \\ H-\vec{V}a \vec{n} \end{array}\right] $$ corresponding to eigenvalue $$\lambda_2=|\vec{V}-a|$$


 * $$R_3 = \left[

\begin{array}{c} n_x \\ un_x \\ vn_x+an_z \\ wn_x-an_y \\ \dfrac{\vec{V}^2}{2}n_x+a(vn_z-wn_y) \end{array}\right] $$ corresponding to eigenvalue $$\lambda_3=|\vec{V}|$$


 * $$R_4 = \left[

\begin{array}{c} n_y \\ un_y-an_z \\ vn_y \\ wn_y+an_x \\ \dfrac{\vec{V}^2}{2}n_y+a(wn_x-un_z) \end{array}\right] $$ corresponding to eigenvalue $$\lambda_3=|\vec{V}|$$


 * $$R_5 = \left[

\begin{array}{c} n_z \\ un_z+an_y \\ vn_z-an_x \\ wn_z \\ \dfrac{\vec{V}^2}{2}n_z+a(un_y-vn_x) \end{array}\right] $$ corresponding to eigenvalue $$\lambda_3=|\vec{V}|$$

$$ P^{-1} = \left[ \begin{array}{c c c c c} \hat{\gamma}\dfrac{\vec{V}^2}{2}+\vec{V}a & -an_x-\hat{\gamma}u & -an_y-\hat{\gamma}v & -an_z-\hat{\gamma}w & \hat{\gamma} \\ \\ \hat{\gamma}\dfrac{\vec{V}^2}{2}-\vec{V}a & an_x-\hat{\gamma}u & an_y-\hat{\gamma}v & an_z-\hat{\gamma}w & \hat{\gamma} \\ \\ 2a(wn_y-vn_z+an_x)-n_x\vec{V}^2\hat{\gamma} & 2un_x\hat{\gamma} &   2vn_x\hat{\gamma}+2a n_z  & 2wn_x\hat{\gamma}-2a n_y & -2n_x\hat{\gamma} \\ \\ 2a(un_z-wn_x+an_y)-n_y\vec{V}^2\hat{\gamma} & 2un_y\hat{\gamma}+2a n_z &  2vn_y\hat{\gamma} & 2wn_y\hat{\gamma}+2a n_x & -2n_y\hat{\gamma} \\ \\ 2a(vn_x-un_y+an_z)-n_z\vec{V}^2\hat{\gamma} & 2un_z\hat{\gamma}+2a n_y &  2vn_z\hat{\gamma}-2a n_x & 2wn_z\hat{\gamma} & -2n_z\hat{\gamma} \\ \end{array}\right]\frac{1}{2a^2} $$