Changes between Version 2 and Version 3 of u/johannjc/scratchpad7

Timestamp:

10/12/15 13:24:17 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad7

@@ -87 +87 @@
 $ \partial_t T = \frac{ T'_0 - T_0}{\Delta t}$
 
-$ \partial_i B_{ij} \partial_j T^{\lambda+1} = \frac{B^{ij}_{\hat{i}} T^{\lambda+1}_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda+1}_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda+1}_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2}$
-
-$ \partial_i B_{ij} \partial_j T^{\lambda}T' = \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}$
+$ \partial_i B_{ij} \partial_j T^{\lambda+1} = \left [ \frac{B^{ij}_{\hat{i}} T^{\lambda+1}_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda+1}_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda+1}_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ] \left ( 1-\delta_{ij} \right )$
+$ + \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda+1}_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda+1}_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda+1}_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda+1}_{\hat{-i}}}{\Delta x^2} \right ] \delta_{ij} $
+
+$ \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 )  $
+$+ \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda}_{\hat{i}}T'_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda}_{0}T'_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda}_{0}T'_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda}_{-hat{i}}T'_{-hat{i}}}{\Delta x^2} \right ] \delta_{ij}$
 
 Using the above definitions, we can write the discretized equation as