User:Arthurv/Notebook

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[edit] 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]

[edit] 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}