Jang Equation In Spherical Symmetry Case
- Jang Equation In Spherical Symmetry Case
Let us assume that the metric is \((\gamma_{rr},\gamma_{\theta\theta},\gamma_{\phi\phi})\), then we calculate the Mean curvature and trace of extrinsic curvature
\[\begin{align}
-H&=\nabla \cdot ( \frac{\gamma^{rr}f_{,r}}{\sqrt{1+\|\nabla f\|^2}})_{,r}\\
&=\frac{1}{\sqrt{\gamma}}(\sqrt{\gamma} \frac{\gamma^{rr}f_{,r}}{\sqrt{1+\gamma^{rr}f_r^2}})_{,r}\\
&=\frac{1}{\sqrt{\gamma}}(\sqrt{\gamma_\theta\gamma_\phi} \frac{\sqrt{\gamma^{rr}}f_{,r}}{\sqrt{1+\gamma^{rr}f^2_{,r}}})_{,r}\\
K_{jang} &=K^r_r+2K^\theta_\theta-\frac{\gamma^{rr}f_r^2K^r_r}{1+\gamma^{rr}f^2_r}
\end{align}
\]
The the Jang's equation in spherical symmetry is
\[\begin{equation}
\boxed{\frac{1}{\sqrt{\gamma}}(\sqrt{\gamma_\theta\gamma_\phi} \frac{\sqrt{\gamma^{rr}}f_{,r}}{\sqrt{1+\gamma^{rr}f^2_{,r}}})_{,r}-K^r_r\frac{\gamma^{rr}f_r^2}{1+\gamma^{rr}f^2_r}+trK=0 }
\end{equation}
\]
In conformal flat matric $$\gamma_{ij}=\phi4(dr2+r2+r2\sin(\theta)^2)$$
\[\begin{equation}
\boxed{
\frac{1}{r^2\phi^6} ( \phi^2r^2 \frac{f_r}{\sqrt{1+\phi^{-4}f_r^2}} )_{,r}-K_r^r \frac{\phi^{-4}f_r^2}{1+\phi^{-4}f_r^2}+trK +\varepsilon f=0
}
\end{equation}
\]
- IMEX for Spherical Symmetry Jang's equation
- Inner 5 points center difference
we talk about the discrete of $\displaystyle \frac{f_x}{\sqrt{1+f_x^2}} $.
\[f^{'}_i=\frac{1}{12h}(f_{i-2}-8f_{i-1}+8f_{i+1}-f_{i+2})
\]
- Out: WENO 5
For the whole part $\displaystyle \phi_x:=(\frac{f_x}{\sqrt{1+f_x^2}})_x $, we use WENO5 as follow
\[ \begin{align}
\phi^{-}_{x,i} &=\frac{1}{12}(-\frac{\Delta^+\phi_{i-2}}{\Delta x}+7\frac{\Delta^{+}\phi_{i-1}}{\Delta x}+7\frac{\Delta^{+}\phi_{i}}{\Delta x}-\frac{\Delta^+ \phi_{i+1}}{\Delta x})\\
&-\Phi^{weno}(\frac{\Delta^-\Delta^+ \phi_{i-2}}{\Delta x},\frac{\Delta^-\Delta^+ \phi_{i-1}}{\Delta x},\frac{\Delta^-\Delta^+ \phi_{i}}{\Delta x},\frac{\Delta^-\Delta^+ \phi_{i+1}}{\Delta x})\\
\phi^{+}_{x,i} &=\frac{1}{12}(-\frac{\Delta^+\phi_{i-2}}{\Delta x}+7\frac{\Delta^{+}\phi_{i-1}}{\Delta x}+7\frac{\Delta^{+}\phi_{i}}{\Delta x}-\frac{\Delta^+ \phi_{i+1}}{\Delta x})\\
&+\Phi^{weno}(\frac{\Delta^-\Delta^+ \phi_{i+2}}{\Delta x},\frac{\Delta^-\Delta^+ \phi_{i+1}}{\Delta x},\frac{\Delta^-\Delta^+ \phi_{i}}{\Delta x},\frac{\Delta^-\Delta^+ \phi_{i-1}}{\Delta x})\\
\end{align}
\]
3.IMEX
\[\begin{align}
f_t &=J(f)+\varepsilon f\\
f_t &=a f_{rr}+J(f)+\varepsilon f-af_{rr}\\
\frac{1}{dt} f^{n+1} -\frac{1}{dt} f^{n} &=aD_2 f^{n+1}+J(f^n) +\varepsilon f^{n+1}-af^n_{rr}\\
(\frac{1}{dt}I-\varepsilon I-aD_2)f^{n+1}&=\frac{1}{dt}f^n+J(f^n)-af^n_{rr}\\
f^{n+1}&=L^{-1}S(f^n)
\end{align}
\]
for the \(D_2\), we use five points center difference
\[f^{''}(x_i)=-\frac{1}{12h^2}(f_{i-2}-16f_{i-1}+30f_i-16f_{i+1}+f_{i+2})
\]
