Changes between Version 10 and Version 11 of u/johannjc/scratchpad4

Timestamp:

09/30/15 17:32:39 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad4

@@ -58 +58 @@
 
 we get expressions for
+
 $\phi' = \frac{\left ( \phi-\psi\lambda \right )}{\psi \left (\lambda+1 \right )} $
 
@@ -82 +83 @@
 We can also write the equation as 
 
-$-T_0 - \alpha_0 T^{\lambda+1}_0 - \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} - \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} = -T'_0 + \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}}$ 
+
+$-T_0 -  \phi' \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 ] = $
+$-T'_0 + \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}}$
 
 where
 
-$\alpha_0 = -\frac{2E_{jj}\Delta t}{\Delta x^2}T_0^\lambda$
+$\alpha_0 = -\frac{2E_{jj}\Delta t}{\Delta x^2}$
 
 $\alpha_{\pm j} = \pm \frac{C_j \Delta t}{2 \Delta x} + \frac{E_{jj}\Delta t}{\Delta x^2}$