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scratchpad4

-              v9
+              v10
 we get expressions for
+$\phi' = \frac{\left ( \phi-\psi\lambda \right )}{\psi \left (\lambda+1 \right )} $
 $B_j =  \frac{\kappa_\parallel \left ( \phi-\psi\lambda \right )}{\lambda+1} \partial_i n b_i b_j $
+$B_j =  \phi' C_j $
 $C_j =   \kappa_\parallel \psi \partial_i n b_i b_j $
 $D_{ij}=\frac{\kappa_\parallel \left ( \phi-\psi\lambda \right )}{\lambda+1} n b_i b_j $
+$D_{ij}=\phi'E_{ij}$
 $E_{ij} =   \kappa_\parallel \psi  n b_i b_j $
 …
 where
 $\alpha_0 = -\frac{1}{\Delta t} -\frac{2E_{jj}}{\Delta x^2}T_0^\lambda$
+$\alpha_0 = -\frac{2E_{jj}\Delta t}{\Delta x^2}T_0^\lambda$
 $\alpha_{\pm j} = \pm C_jT^\lambda_{\pm \hat{j}} + \frac{E_{jj}}{\Delta x^2}T_{\pm \hat{j}}^\lambda$
+$\alpha_{\pm j} = \pm \frac{C_j \Delta t}{2 \Delta x} + \frac{E_{jj}\Delta t}{\Delta x^2}$
+$\alpha_{\pm i, \pm j} = \pm \pm \frac{E_{ij}}{4 \Delta x^2}T^\lambda_{\pm \hat{i} \pm \hat{j}}$
+$\beta = -\frac{T_0}{\Delta t} \pm \frac{B_j}{2 \Delta x} T^{\lambda+1}_{\pm \hat{j}} \pm \pm \frac{D_{ij}}{4 \Delta x^2} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}}\left ( 1 - \delta_{ij} \right )  + \frac{D_{jj}}{\Delta x^2} T^{\lambda+1}_{\pm \hat{j}} - 2 \frac{D_{jj}}{\Delta x^2}T^{\lambda+1}_0$
+$\alpha_{\pm i, \pm j} = \pm \pm \frac{E_{ij}\Delta t}{4 \Delta x^2}$