Changes between Version 1 and Version 2 of u/johannjc/scratchpad6

Timestamp:

10/09/15 14:00:47 (10 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad6

@@ -68 +68 @@
 This makes the matrix equation for $\Delta T'_{\vec{k}}$
 
-$\left [  \frac{\delta_{\vec{k}}^{\vec{k'}}}{\Delta t} -  \phi \left ( \lambda + 1 \right ) \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \delta^{\vec{k'}}_{\vec{k}}\left ( \psi T_{\vec{k'}} + \phi T'_{\vec{k'}} \right ) ^{\lambda} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \delta^{\vec{k'}\pm \hat{j}}_{ \vec{k}} \left (\psi T_{\vec{k'}\pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{j}} \right )^{\lambda} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \delta^{\vec{k'}\pm \hat{i} \pm \hat{j}}_{\vec{k}} \left ( \psi T_{\vec{k'}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \right ] \right ] \Delta T'_{\vec{k}}$
+$\left [  \frac{\delta_{\vec{k}}^{\vec{k'}}}{\Delta t} -   \displaystyle \sum_{\parallel, \perp} \phi \left ( \lambda + 1 \right ) \left [ \alpha_0 \delta^{\vec{k'}}_{\vec{k}}\left ( \psi T_{\vec{k'}} + \phi T'_{\vec{k'}} \right ) ^{\lambda} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \delta^{\vec{k'}\pm \hat{j}}_{ \vec{k}} \left (\psi T_{\vec{k'}\pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{j}} \right )^{\lambda} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \delta^{\vec{k'}\pm \hat{i} \pm \hat{j}}_{\vec{k}} \left ( \psi T_{\vec{k'}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \right ] \right ] \Delta T'_{\vec{k}}$
 $= -\Xi_{\vec{k'}}(\vec{T'}) $
 
@@ -74 +74 @@
 
 
-$ \frac{\Delta T'_{\vec{k'}}}{\Delta t} -  \phi \left ( \lambda + 1 \right ) \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \left ( \psi T_{\vec{k'}} + \phi T'_{\vec{k'}} \right ) ^{\lambda} \Delta T'_{\vec{k'}} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\vec{k'}\pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k'} \pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j}  \left ( \psi T_{\vec{k'}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k'}} \right ] =-\Xi_{\vec{k'}}(\vec{T'})$
-
-This gives our first correction as 
-
-$T''_0 = T'_0 - \frac{T_0' - T_0 -  \Delta t \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \left ( \psi T_0 + \phi T'_0 \right ) ^{\lambda+1} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\pm \hat{j}} + \phi T'_{\pm \hat{j}} \right )^{\lambda+1} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \left ( \psi T_{\pm \hat{i} \pm \hat{j}} + \phi T'_{\pm \hat{i} \pm \hat{j}} \right )^{\lambda+1} \right ]}{1 -  \Delta t \phi \left ( \lambda + 1 \right ) \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \left ( \psi T_0 + \phi T'_0 \right ) ^{\lambda} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\pm \hat{j}} + \phi T'_{\pm \hat{j}} \right )^{\lambda} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \left ( \psi T_{\pm \hat{i} \pm \hat{j}} + \phi T'_{\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \right ]}$
+$ \frac{\Delta T'_{\vec{k'}}}{\Delta t} -  \displaystyle \sum_{\parallel, \perp} \phi \left ( \lambda + 1 \right )  $
+$\left [ \alpha_0 \left ( \psi T_{\vec{k'}} + \phi T'_{\vec{k'}} \right ) ^{\lambda} \Delta T'_{\vec{k'}} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\vec{k'}\pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k'} \pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j}  \left ( \psi T_{\vec{k'}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k'}\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k'}\pm \hat{i} \pm \hat{j}} \right ]$
+$ =-\Xi_{\vec{k'}}(\vec{T'})$
 
 
 or
 
-$T''_0 = \frac{T_0 +  \Delta t \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \left ( \psi T_0 + \phi T'_0 \right ) ^{\lambda}\left ( \psi T_0 - \phi \lambda T'_0 \right ) +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\pm \hat{j}} + \phi T'_{\pm \hat{j}} \right )^{\lambda} \left (\psi T_{\pm \hat{j}} - \phi \lambda T'_{\pm \hat{j}} \right ) +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \left ( \psi T_{\pm \hat{i} \pm \hat{j}} + \phi T'_{\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \left ( \psi T_{\pm \hat{i} \pm \hat{j}} - \phi \lambda T'_{\pm \hat{i} \pm \hat{j}} \right ) \right ]}{1 -  \Delta t \phi \left ( \lambda + 1 \right ) \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \left ( \psi T_0 + \phi T'_0 \right ) ^{\lambda} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\pm \hat{j}} + \phi T'_{\pm \hat{j}} \right )^{\lambda} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \left ( \psi T_{\pm \hat{i} \pm \hat{j}} + \phi T'_{\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \right ]}$
+$ \Delta T'_{\vec{k}}-  \Delta t \phi \left ( \lambda + 1 \right ) \displaystyle \sum_{\parallel, \perp}  $
+$\left [ \alpha_0 \left ( \psi T_{\vec{k}} + \phi T'_{\vec{k}} \right ) ^{\lambda} \Delta T'_{\vec{k}} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\vec{k}\pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k} \pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j}  \left ( \psi T_{\vec{k}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k}\pm \hat{i} \pm \hat{j}} \right ]$
+
+$ =  -T'_{\vec{k}} + T_{\vec{k}} +  \Delta t \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \left ( \psi T_{\vec{k}} + \phi T'_{\vec{k}} \right ) ^{\lambda+1} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\vec{k}\pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{j}} \right )^{\lambda+1} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \left ( \psi T_{\vec{k}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{i} \pm \hat{j}} \right )^{\lambda+1} \right ]$
+
+
+or we can write it as 
+
+
+$ T''_{\vec{k}} = T'_{\vec{k}} + \Delta T'_{\vec{k}}  =  T_{\vec{k}} +  \Delta t \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0 \left ( \psi T_{\vec{k}} + \phi T'_{\vec{k}} \right ) ^{\lambda+1} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\vec{k}\pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{j}} \right )^{\lambda+1} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} \left ( \psi T_{\vec{k}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{i} \pm \hat{j}} \right )^{\lambda+1} \right ] $
+$  +\Delta t \displaystyle \sum_{\parallel, \perp} \phi \left ( \lambda + 1 \right )  $
+$\left [ \alpha_0 \left ( \psi T_{\vec{k}} + \phi T'_{\vec{k}} \right ) ^{\lambda} \Delta T'_{\vec{k}} +  \displaystyle \sum_{\pm j} \alpha_{\pm j} \left (\psi T_{\vec{k}\pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k} \pm \hat{j}} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j}  \left ( \psi T_{\vec{k}\pm \hat{i} \pm \hat{j}} + \phi T'_{\vec{k}\pm \hat{i} \pm \hat{j}} \right )^{\lambda} \Delta T'_{\vec{k}\pm \hat{i} \pm \hat{j}} \right ]$
+
+
+Now if we start with $T'=T$ and set $\Delta T' = T''-T$ we end up with
+
+$ T''_{\vec{k}} =  T_{\vec{k}} +  \Delta t \displaystyle \sum_{\parallel, \perp}  \left [ \alpha_0  T_{\vec{k}}^{\lambda+1} +  \displaystyle \sum_{\pm j} \alpha_{\pm j}T_{\vec{k}\pm \hat{j}}^{\lambda+1} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T_{\vec{k}\pm \hat{i} \pm \hat{j}}^{\lambda+1} \right ] $
+$  +\Delta t \displaystyle \sum_{\parallel, \perp}  \phi \left ( \lambda + 1 \right ) $
+$\left [ \alpha_0 T_{\vec{k}}^{\lambda} \left(T''_{\vec{k}} - T_{\vec{k}} \right ) +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T_{\vec{k}\pm \hat{j}}^{\lambda}  \left(T''_{\vec{k}\pm \hat{j}} - T_{\vec{k}\pm \hat{j}} \right )  +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T_{\vec{k}\pm \hat{i} \pm \hat{j}}^{\lambda} \left ( T''_{\vec{k}\pm \hat{i} \pm \hat{j}} - T_{\vec{k}\pm \hat{i} \pm \hat{j}} \right )  \right ]$
+
+which can be rearranged as
+
+
+$ T''_{\vec{k}} -  \Delta t  \displaystyle \sum_{\parallel, \perp}  \phi \left ( \lambda + 1 \right ) $
+$\left [ \alpha_0 T_{\vec{k}}^{\lambda} T''_{\vec{k}}  +  \displaystyle \sum_{\pm j} \alpha_{\pm j} T_{\vec{k}\pm \hat{j}}^{\lambda} T''_{\vec{k}\pm \hat{j}}  +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T_{\vec{k}\pm \hat{i} \pm \hat{j}}^{\lambda}T''_{\vec{k}\pm \hat{i} \pm \hat{j}} \right ] $
+$=  T_{\vec{k}} +  \Delta t \displaystyle \sum_{\parallel, \perp}  \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )  \left [ \alpha_0  T_{\vec{k}}^{\lambda+1} +  \displaystyle \sum_{\pm j} \alpha_{\pm j}T_{\vec{k}\pm \hat{j}}^{\lambda+1} +  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T_{\vec{k}\pm \hat{i} \pm \hat{j}}^{\lambda+1} \right ] $
+
+
+which is the linearized implicit equation we are solving.  Not surprisingly, a single-step newton method is what we are doing by Taylor expanding $T*^{\lambda+1}$.