Changes between Version 11 and Version 12 of u/johannjc/scratchpad7

Timestamp:

10/12/15 15:47:11 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad7

@@ -91 +91 @@
 
 $ \partial_i B_{ij} \partial_j T^{\lambda}T' =  \left [ \frac{B^{ij}_{\hat{i}} T^{\lambda}_{\hat{i}+\hat{j}}T'_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda}_{\hat{i}-\hat{j}}T'_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda}_{-\hat{i}+\hat{j}}T'_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda}_{-\hat{i}-\hat{j}}T'_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ]  \left ( 1-\delta_{ij} \right )  $
-$+ \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda}_{\hat{i}}T'_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda}_{0}T'_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda}_{0}T'_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda}_{-hat{i}}T'_{-hat{i}}}{\Delta x^2} \right ] \delta_{ij}$
+$+ \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda}_{\hat{i}}T'_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda}_{0}T'_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda}_{0}T'_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2} \right ] \delta_{ij}$
 
 Using the above definitions, we can write the discretized equation as 
@@ -207 +207 @@
 For the Explicit case, we just set $\phi = 0$ which sets $D = 0$ and $C=1$ and we have
 
-$T_0 +  \Delta t \displaystyle \sum_{\parallel, \perp} \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = T'_0$
+$T_0 +  \Delta t \displaystyle \sum_{\parallel, \perp} A \left [ \alpha_0 T^{\lambda+1}_0 +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = T'_0$
 
 === Explicit time stepping ===
@@ -213 +213 @@
 For the explicit scheme to be stable, we must have
 
-$\Delta t < \frac{1}{2\displaystyle \max_{\parallel, \perp, i} \left [ | \alpha_i T_i^{\lambda} |  \right ]}$
+$\Delta t < \frac{1}{2\displaystyle \max_{\parallel, \perp, i} A \left [ | \alpha_i T_i^{\lambda} |  \right ]}$