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 \[\gamma_{ij}=\phi^4(dr^2+r^2+r^2\sin(\theta)^2)\]
  \[ \begin{equation} \boxed{ \frac{1}{r^2\phi^6} ( \phi^2r^2 \frac{f_r}{\sqrt{1+\phi^{-4}f_r^2}} )_{,r}-K_r^r \frac{\phi^{-4}f_r^2}{1+\phi^{-4}f_r^2}+trK +\varepsilon f=0 } \end{equation} \]
• IMEX for Spherical Symmetry Jang's equation

1. Inner 5 points center difference
   we talk about the discrete of $\displaystyle \frac{f_x}{\sqrt{1+f_x^2}} $.
   \[f^{'}_i=\frac{1}{12h}(f_{i-2}-8f_{i-1}+8f_{i+1}-f_{i+2})\]
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
   \[f^{''}(x_i)=-\frac{1}{12h^2}(f_{i-2}-16f_{i-1}+30f_i-16f_{i+1}+f_{i+2})\]