Changes between Version 2 and Version 3 of u/johannjc/scratchpad4

Timestamp:

09/30/15 16:42:07 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad4

@@ -55 +55 @@
 Now if we write the equation as 
 
-$\partial_t T = B_i \partial_j T^{\lambda+1} + C_j \partial_j T^\lambda T' + D_{ij} \partial_i\partial_j T^{\lambda+1} + E_{ij} \partial_i \partial_j T^\lambda T'$
+$\partial_t T = B_j \partial_j T^{\lambda+1} + C_j \partial_j T^\lambda T' + D_{ij} \partial_i\partial_j T^{\lambda+1} + E_{ij} \partial_i \partial_j T^\lambda T'$
 
 we get expressions for
+
+$A = \frac{1}{\Delta t}$
 
 $B_j =  \frac{\kappa_\parallel \left ( \phi-\psi\lambda \right )}{\lambda+1} \partial_i n b_i b_j $
@@ -69 +71 @@
 We then need expressions for
 
-$ \partial_t T = \frac{T'_0 - T_0}{\Delta t}$
+$ \partial_t T = A \left ( T'_0 - T_0 \right )$
+
 $ \partial_j T^{\lambda+1} = \frac{T^{\lambda+1}_{\hat{j}} - T^{\lambda+1}_{-\hat{j}}}{2 \Delta x}$
 
@@ -80 +83 @@
 We can also write the equation as 
 
-$\alpha_0 T'_0 + \displaystyle \sum_{\pm i} \alpha_{\pm i} T'_{\pm \hat{i}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T'_{\pm \hat{i} \pm \hat{j}} = \beta$ 
+$\beta = \alpha_0 T'_0 + \displaystyle \sum_{\pm i} \alpha_{\pm i} T'_{\pm \hat{i}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T'_{\pm \hat{i} \pm \hat{j}}$ 
 
 and then work out the coefficients for the matrix equation
 
+$\alpha_0 = -A -2E_{ii}T_0^\lambda = n \kappa_\parallel \psi$ 
+
+$\alpha_{\pm i} = \pm C_iT^\lambda_{\pm hat{i}} + E_{ii}T_{\pm \hat{i}}^\lambda$
+