Changes between Version 14 and Version 15 of u/johannjc/scratchpad

Timestamp:

01/10/14 14:46:10 (11 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -1 @@
-== Self-gravitating equilibrium profile using density profile and external pressure ==
+[[latex($\rho c_v \frac{\partial T}{\partial t} = \nabla \left ( \kappa T^{n} \nabla T \right )$)]]
 
-Mathematically, we are solving the equation of hydrostatic equilibrium 
+[[latex($\rho c_v \frac{\partial T}{\partial t} =\nabla^2 \left ( \frac{1}{n+1} \kappa T^{n+1} \right )$)]]
 
-[[latex($\nabla P(r) = -\rho(r) \nabla \phi(r)$)]] 
-
-subject to the constraint that  [[latex($P(R)=P_0$)]].  
-
-Since [[latex($\phi(r) = -\frac{G M(r)}{r}$)]] we can rewrite the equation for HSE as 
-{{{
-#!latex
-\begin{equation}
-\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^2}
-\end{equation}
-}}}
+[[latex($\rho c_v \frac{\partial T}{\partial t} =\nabla^2 \left ( \frac{1}{n+1} \kappa T^{n} T \right )$)]]
 
 
- and we can express the enclosed mass as 
+[[latex($\rho_i c_v \left ( T_i^{j+1}-T_i^j \right ) = \frac{\kappa \Delta t}{\Delta x^2}\frac{1}{n+1} \left (T_{i+1}^{j^n}  T_{i+1}^{j+1} - 2 T_{i}^{j^n}  T_{i}^{j+1} + T_{i-1}^{j^n}  T_{i-1}^{j+1} \right )$)]]
 
-{{{
-#!latex
-\setcounter{equation}{1}
-\begin{equation}
-M(r)=4\pi\displaystyle\int_0^r{\rho(r') r'^2dr'}
-\end{equation}
-}}}
+[[latex($\alpha = \frac{\kappa \Delta t}{\Delta x^2 \left ( n+1 \right ) }$)]]
 
-Now we can discretize equation 2 as
-
-{{{
-#!latex
-\setcounter{equation}{2}
-\begin{equation}
-M_{i+1}=M_i + 4\pi \rho_i r_i^2\Delta r_i
-\end{equation}
-}}}
-
-where [[latex($\Delta r_j = r_{j+1}-r_j$)]]
-
-and equation 1 as
-
-{{{
-#!latex
-\setcounter{equation}{3}
-\begin{equation}
-P_{i+1}=P_i-\Delta r_i \frac{G M_i \rho_i}{r_i^2}
-\end{equation}
-}}}
+[[latex($ \left (\rho_i c_v + 2 \alpha T_i^{j^n} \right ) T_i^{j+1} - \alpha T_{i+1}^{j^n} T_{i+1}^{j+1} - \alpha T_{i-1}^{j^n} T_{i-1}^{j+1} =  \rho_i c_v T_i^j$)]]
 
 
-but a taylor series expansion shows that the error at each step in the integration goes as 
-
-[[latex($\frac{\Delta r_j}{r_j} + \frac{\Delta \rho_j}{\rho_j} + ...$)]]
-
-so we need to limit out point spacing - or improve the accuracy of our discretization.
-
-If we apply linear interpolation to our density profile, we can discretize equation 2 as
-
-{{{
-#!latex
-\setcounter{equation}{5}
-\begin{equation}
-M_i=M(r_i)=4\pi\displaystyle\sum_{j=0}^i{\rho_j r_j^2\Delta r_j \left ( 1 + \frac{\Delta r_j}{r_j} + \frac{\Delta \rho}{2\rho} + \frac{\Delta r_j^2}{3 r_j^2} + \frac{2\Delta \rho \Delta r_j}{3 \rho r_j} + \frac{\Delta \rho \Delta r_j^2}{4 \rho r_j^2}\right )}
-\end{equation}
-}}}
-
-and equation 4 becomes a bit of a monster - and we have to worry about limits as r -> 0.
-{{{
-if (r0 == 0d0) then
-    dp=drho*((pi*drho*dr**4)/4d0 + (4d0*pi*rho*dr**3)/9d0) + (rho*(pi*drho*dr**3 + 2d0*pi*rho*dr**2))/3d0
-else
-    epsilon=log(r0/(r0+dr))
-    dp=((-36d0*drho*r0)*epsilon*M0*dr + (-36d0*drho*r0**2)*epsilon*M0 + (- 12d0*pi*drho**2*r0**5 + 48d0*pi*rho*drho*r0**4)*epsilon*dr + (-12d0*pi*drho**2*r0**6 + 48d0*pi*rho*drho*r0**5)*epsilon + (36d0*rho - 36d0*drho*r0)*M0*dr + (9d0*pi*drho**2*r0)*dr**5 + (17d0*pi*drho**2*r0**2 + 28d0*pi*rho*drho*r0)*dr**4 + (2d0*pi*drho**2*r0**3 + 64d0*pi*drho*r0**2*rho + 24d0*pi*r0*rho**2)*dr**3 + (- 6d0*pi*drho**2*r0**4 + 24d0*pi*drho*r0**3*rho + 72d0*pi*r0**2*rho**2)*dr**2 +(- 12d0*pi*drho**2*r0**5 + 48d0*pi*rho*drho*r0**4)*dr)/(36d0*r0**2 + 36d0*dr*r0)
-end if
-data(i,i_Press)=data(i+1,i_Press)+ScaleGrav*dp
-}}}
+Now to make the scheme 2nd order in time we can replace [[latex($T_i^{j+1}$)]] with [[latex($\frac{T_i^{j}+T_i^{j+1}}{2}$)]]
 
 
+[[latex($ \left (\rho_i c_v + 2 \beta T_i^{j^n} \right ) T_i^{j+1} - \beta T_{i+1}^{j^n} T_{i+1}^{j+1} - \beta T_{i-1}^{j^n} T_{i-1}^{j+1} =  \rho_i c_v T_i^j -  2 \beta T_i^{j^n} T_i^{j} + \beta T_{i+1}^{j^n} T_{i+1}^{j} + \beta T_{i-1}^{j^n} T_{i-1}^{j} $)]]
+
+
+Which simplifies to:
+
+[[latex($\left (\rho_i c_v + 2 \beta T_i^{j^n} \right ) T_i^{j+1} - \beta T_{i+1}^{j^n} T_{i+1}^{j+1} - \beta T_{i-1}^{j^n} T_{i-1}^{j+1} =  \rho_i c_v T_i^j -  2 \beta T_i^{j^{n+1}} + \beta T_{i+1}^{j^{n+1}} + \beta T_{i-1}^{j^{n+1}}$)]]
+
+