Changes between Version 4 and Version 5 of u/johannjc/scratchpad5

Timestamp:

10/01/15 10:50:26 (11 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad5

@@ -27 +27 @@
 or 
 
-$\partial_t T = \partial_i n b_i \frac{\kappa_\parallel}{\lambda_\parallel+1} b_j \partial_j T^{\lambda_\parallel+1} +   \partial_i n \left ( \delta_{ij} - b_i b_j \right ) \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_j T^{\lambda_\parallel+1} $
-
-or 
-
-$\partial_t T = \frac{\kappa_\parallel}{\lambda_\parallel+1}  \partial_i n b_i  b_j \partial_j T^{\lambda_\parallel+1} +   \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_i n \left ( \delta_{ij} - b_i b_j \right )  \partial_j T^{\lambda_\parallel+1} $
+$\partial_t T = \partial_i n b_i \frac{\kappa_\parallel}{\lambda_\parallel+1} b_j \partial_j T^{\lambda_\parallel+1} +   \partial_i n \left ( \delta_{ij} - b_i b_j \right ) \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_j T^{\lambda_\perp+1} $
+
+or 
+
+$\partial_t T = \frac{\kappa_\parallel}{\lambda_\parallel+1}  \partial_i n b_i  b_j \partial_j T^{\lambda_\parallel+1} +   \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_i n \left ( \delta_{ij} - b_i b_j \right )  \partial_j T^{\lambda_\perp +1} $
+
+Now both of these terms are of the form
+
+$ A \partial_i  B_{ij} \partial_j T^{\lambda+1}$
+
+Now to solve this implicitly, we need to replace $T$ with $T_*$ where 
+
+$T_* = T + \phi ( T' - T ) = (1-\phi) T + \phi T'$
+
+Note for Backward Euler, $\phi = 1$ and for Crank Nicholson, $\phi = 1/2$
+
+$ A \partial_i  B_{ij} \partial_j T_*^{\lambda+1}$
+
+so we have
+
+$A \partial_i  B_{ij} \partial_j \left ((1-\phi) T + \phi T' \right )^{\lambda+1}$
+
+Now to solve this using a linear system, we need to linearize terms involving $T'$
+
+So we need to Taylor expand about $T$
+
+$\left ((1-\phi) T + \phi T' \right )^{\lambda+1} \approx T^{\lambda + 1} + \left ( \lambda + 1 \right ) T^{\lambda} \phi \left ( T' - T \right ) $
+
+or
+
+$\left ((1-\phi) T + \phi T' \right )^{\lambda+1} \approx \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T'$
+
+Now we could also have expanded $T_*^{\lambda+1}$ and then plug in $T_*=\left ( 1 - \phi \right ) T + \phi T'$.  In that case we also get
+
+
+$T_*^{\lambda + 1} \approx T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \left ( T_{*} - T \right ) =  T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \phi \left ( T' - T \right )$
+$= \left(1 - \phi \left ( \lambda + 1\right ) \right ) T^{\lambda + 1}+  \phi \left ( \lambda + 1 \right ) T^{\lambda} T' $
+
+
+So we have
+
+$A \partial_i  B_{ij} \partial_j \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ]$
+
+$A \partial_i  B_{ij} \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right ) \partial_j T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) \partial_j T^{\lambda} T' \right ]$
+
+$A \partial_i  B_{ij} \left [ C_\lambda  \partial_j T^{\lambda + 1} + D_\lambda  \partial_j T^{\lambda} T' \right ]$
+
+where $ C_\lambda= \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )$ and $ D_\lambda = \phi \left ( \lambda + 1 \right )$
+
+Now we can expand the derivatives and get
+
+$A \left ( \partial_i  B_{ij} \right ) \left ( C_\lambda  \partial_j T^{\lambda + 1} + D_\lambda  \partial_j T^{\lambda} T' \right ) + A B_{ij} \left ( C_\lambda  \partial_i \partial_j T^{\lambda + 1} + D_\lambda  \partial_i \partial_j T^{\lambda} T' \right ) $
+
+