Changes between Version 5 and Version 6 of u/johannjc/scratchpad7

Timestamp:

10/12/15 14:10:01 (11 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad7

@@ -96 +96 @@
 
 
-$T_0 + \Delta t \displaystyle \sum_{\parallel, \perp} C \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
-$T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} D \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
+$T_0 + \Delta t \displaystyle \sum_{\parallel, \perp} AC \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
+$T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
 
 where
@@ -105 +105 @@
 $\alpha_{\pm i} = \frac{B^{ii}_{\pm \hat{i}}}{\Delta x^2}$
 
-$\alpha_{\pm i, \pm j} = \pm \pm \frac{B^{ij}_{\pm \hat{i}}}{4 \Delta x^2} $ where the $\pm$ in $B_{\pm \hat{i}}$ corresponds to the $\pm i$ term.
+$\alpha_{\pm i, \pm j} =  \pm \pm \frac{B^{ij}_{\pm \hat{i}}}{4 \Delta x^2} $ where the $\pm$ in $B_{\pm \hat{i}}$ corresponds to the $\pm i$ term.
+
+
+Also note, that the indexing of the temperatures is commutative, and that all of the diagonal temperature terms will have two contributions.  One from $\alpha \pm i \pm j$ and one from $\alpha \pm j \pm i$.  We can rewrite
+
+$\displaystyle \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} = \sum_{\pm i, \pm j,i < j} \left ( \alpha_{\pm i, \pm j} + \alpha_{\pm j, \pm i} \right ) T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} = \sum_{\pm i, \pm j,i < j} \alpha*_{\pm i, \pm j}T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}}$
+
+where
+
+$\alpha*_{\pm i \pm j} = \pm \pm \frac{B^{ij}_{\pm \hat{i}}+B^{ji}_{\pm \hat{j}}}{4 \Delta x^2}$
 
 
@@ -113 +122 @@
 || $A$ || $\frac{\kappa_\parallel}{\rho c_v  \left ( \lambda_\parallel + 1 \right ) }$ || $\frac{\kappa_\perp}{\rho c_v  \left (\lambda_\perp + 1\right )}$ ||
 || $B_{ij} $ || $b_i b_j$ || $n^2 B^{-2} b_i b_j \left ( 1 - \delta_{ij} \right )$ ||
-|| $C$ ||  $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ ||  $\left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ ||
-|| $D$ ||  $\phi \left ( \lambda_\parallel + 1 \right )$ ||  $\phi \left ( \lambda_\perp + 1 \right ) $||
-
-And then using those we can calculate
-
-|| $E_j$ || $A \left ( \partial_i  B_{ij} \right )$ ||
-|| $F_{ij}$ || $A B_{ij}$ ||
-||$\alpha_0 $ ||$ -\frac{2F_{ij}\delta_{ij} }{\Delta x^2}$ ||
-||$\alpha_{\pm j}$ || $ \pm \frac{E_j }{2 \Delta x} + \frac{F_{jj}\Delta t}{\Delta x^2}$ ||
-||$\alpha_{\pm i, \pm j}$ || $ \pm \pm \frac{F_{ij}}{4 \Delta x^2}$ ||
+|| $C$ ||  $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ ||  $ \left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ ||
+|| $D$ ||  $\phi \left ( \lambda_\parallel + 1 \right )$ ||  $ \phi \left ( \lambda_\perp + 1 \right ) $||
+
+||$\alpha_0$ || $- \displaystyle \sum_{i} \frac{B^{ii}_{\hat{i}} + B^{ii}_{-\hat{i}}}{\Delta x^2}$ ||
+||$\alpha_{\pm i}$ || $ \frac{B^{ii}_{\pm \hat{i}}}{\Delta x^2}$ ||
+|| $\alpha*_{\pm i \pm j}$ || $ \pm \pm \frac{B^{ij}_{\pm \hat{i}}+B^{ji}_{\pm \hat{j}}}{4 \Delta x^2}$ ||