Changes between Version 5 and Version 6 of u/johannjc/scratchpad5

Timestamp:

10/01/15 11:09:53 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad5

@@ -78 +78 @@
 Now we can expand the derivatives and get
 
-$A \left ( \partial_i  B_{ij} \right ) \left ( C_\lambda  \partial_j T^{\lambda + 1} + D_\lambda  \partial_j T^{\lambda} T' \right ) + A B_{ij} \left ( C_\lambda  \partial_i \partial_j T^{\lambda + 1} + D_\lambda  \partial_i \partial_j T^{\lambda} T' \right ) $
+$A \left ( \partial_i  B_{ij} \right ) \left ( C  \partial_j T^{\lambda + 1} + D  \partial_j T^{\lambda} T' \right ) + A B_{ij} \left ( C \partial_i \partial_j T^{\lambda + 1} + D  \partial_i \partial_j T^{\lambda} T' \right ) $
+
+and the whole equation is then
+
+$\partial_t T = \displaystyle \sum_{\parallel,\perp}{A \left ( \partial_i  B_{ij} \right ) \left ( C  \partial_j T^{\lambda + 1} + D  \partial_j T^{\lambda} T' \right ) + A B_{ij} \left ( C  \partial_i \partial_j T^{\lambda + 1} + D  \partial_i \partial_j T^{\lambda} T' \right )} $
+
+where $A$, $B$, $C$, $D$, and $\lambda$ are functions of $\perp,\parallel$
+
+We can also write this as 
+
+$\partial_t T = \displaystyle \sum_{\parallel,\perp}{E \partial_j T^{\lambda + 1} + F \partial_j T^{\lambda} T' + G  \partial_i \partial_j T^{\lambda + 1} + H \partial_i \partial_j T^{\lambda} T'} $
+
+where
+
+$E = A \left ( \partial_i  B_{ij} \right ) C$
+
+$F =  A \left ( \partial_i  B_{ij} \right ) D$
+
+$G = A B_{ij} C$
+
+$H = A B_{ij} D$
 
 
-
-
-Let's first just consider the $\chi_\parallel$ term.
-
-$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \kappa_\parallel T^\lambda \left ( \hat{b} \cdot \nabla T \right )\right ]$
-
-We need a way to write this implicitly but we need it to also be linear in $T_{*}$
-
-$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \kappa_\parallel T_{*}^\lambda \left ( \hat{b} \cdot \nabla T_{*} \right )\right ]$
-
-We could Taylor expand $T^\lambda_{*}=T^\lambda + \lambda T^{\lambda-1} \left ( T_{*}-T \right ) = \left ( 1-\lambda \right ) T^\lambda + \lambda T^{\lambda-1} T_{*}$
-
-but then we would still have a non-linear term like $\lambda T^{\lambda-1} T_{*} \nabla T_{*}$ 
-
-We could write this as $\lambda T^{\lambda-1} 1/2 \nabla T_{*}^2$ and Taylor expand again to get $\lambda T^{\lambda-1} 1/2 \nabla \left ( - T + 2 TT_{*} \right )$
-
-but we've now done a Taylor expansion on a Taylor expansion...  
-
-
-Alternatively, we can rewrite the diffusion equation
-
-$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{\kappa_\parallel}{\lambda+1} \left ( \hat{b} \cdot \nabla T_*^{\lambda+1} \right )\right ]$
-
-
-
-and then perform a single Taylor expansion
-
-$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{\kappa_\parallel}{\lambda+1} \left ( \hat{b} \cdot \nabla \left (-\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}T_{*} \right )  \right ) \right ]$
-
-Switching to Einstein notation, we have
-
-$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( -\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}T_{*} \right )  $
-
-$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( -\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}T_{*} \right )  $
-
-Let's also take a moment to write $T_* = \phi T + \psi T'$  where $T'$ is the new temperature, and $\phi + \psi = 1$.  Backward Euler would have $\phi=0$ and $\psi=1$ where Crank-Nicholson would have $\phi=\psi=1/2$
-
-$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( -\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}\left ( \phi T + \psi T' \right ) \right )  $
-
-
-$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( \left ( \phi - \psi \lambda \right ) T^{\lambda+1} + \psi \left ( \lambda + 1 \right ) T^{\lambda} T' \right )   $
-
-
-$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \left [ \left ( \partial_i n b_i b_j \right )  \partial j \left ( \left ( \phi - \psi \lambda \right ) T^{\lambda+1} + \psi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ) +  n b_i b_j \left ( \left ( \phi - \psi \lambda \right ) \partial_i \partial_j T^{\lambda+1} + \psi \left ( \lambda + 1 \right ) \partial_i \partial_j T^{\lambda} T' \right ) \right ]$
-
-
-Now if we write the equation as 
-
-$\partial_t T = B_j \partial_j T^{\lambda+1} + C_j \partial_j T^\lambda T' + D_{ij} \partial_i\partial_j T^{\lambda+1} + E_{ij} \partial_i \partial_j T^\lambda T'$
-
-we get expressions for
-
-$\phi' = \frac{\left ( \phi-\psi\lambda \right )}{\psi \left (\lambda+1 \right )} $
-
-$B_j =  \phi' C_j $   
-
-$C_j =   \kappa_\parallel \psi \partial_i n b_i b_j $
-
-$D_{ij}=\phi'E_{ij}$
-
-$E_{ij} =   \kappa_\parallel \psi  n b_i b_j $
 
 We then need expressions for
@@ -153 +112 @@
 $\partial_i\partial_j T^{\lambda+1} = \frac{T^{\lambda+1}_{\hat{i}+\hat{j}} - T^{\lambda+1}_{\hat{i}-\hat{j}}-T^{\lambda+1}_{-\hat{i}+\hat{j}} + T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2}\left(1-\delta_{ij}\right ) +  \frac{T^{\lambda+1}_{\hat{i}} - 2T^{\lambda+1}_{0} + T^{\lambda+1}_{-\hat{i}}}{\Delta x^2}\delta_{ij}$
 
-$\partial_i\partial_j T^{\lambda}T' = \frac{T^{\lambda}_{\hat{i}+\hat{j}}T'_{\hat{i}+\hat{j}} - T^{\lambda}_{\hat{i}-\hat{j}}T'_{\hat{i}-\hat{j}}-T^{\lambda}_{-\hat{i}+\hat{j}}T'_{-\hat{i}+\hat{j}} + T^{\lambda}_{-\hat{i}-\hat{j}}T'_{-\hat{i}-\hat{j}}}{4 \Delta x^2}\left(1-\delta_{ij}\right ) +  \frac{T^{\lambda}_{\hat{i}}T'_{\hat{i}} - 2T^{\lambda}_{0}T'_{0} + T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2}\delta_{ij}$
+$\partial_i\partial_j T^{\lambda}T' = \frac{T^{\lambda}_{\hat{i}+\hat{j}}T'_{\hat{i}+\hat{j}} - T^{\lambda}_{\hat{i}-\hat{j}}T'_{\hat{i}-\hat{j}}-T^{\lambda}_{-\hat{i}+\hat{j}}T'_{-\hat{i}+\hat{j}} + T^{\lambda}_{-\hat{i}-\hat{j}}T'_{-\hat{i}-\hat{j}}}{4 \Delta x^2}\left(1-\delta_{ij}\right ) $ $ + \frac{T^{\lambda}_{\hat{i}}T'_{\hat{i}} - 2T^{\lambda}_{0}T'_{0} + T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2}\delta_{ij}$
+
 
 We can also write the equation as