Changes between Version 7 and Version 8 of ThermalConduction

Timestamp:

08/15/16 13:24:22 (8 years ago)

Author:

Jonathan

Comment:

—

ThermalConduction

@@ -174 +174 @@
 So in summary we have
 
-|| $A$ || $\frac{\kappa}{\rho c_v  \left ( \lambda + 1 \right ) }$ || 
+|| $A$ || $\frac{1}{\rho c_v  \left ( \lambda + 1 \right ) }$ || 
 || $B_{ij} $ || $\kappa \delta{ij}$||
 || $C$ ||  $\left ( 1 - \phi \left ( \lambda+ 1 \right ) \right )$ ||
@@ -182 +182 @@
 
 ||$\alpha_0$ || $- \displaystyle \sum_{i} \frac{B_{\hat{i}} + B_{-\hat{i}}}{\Delta x^2}$ ||
-||$\alpha_{\pm i}$ || $ \frac{B_{\pm \hat{i}}}{\Delta x^2}$ ||
+||$\alpha_{\pm i}$ || $ \frac{\kappa_{\pm \hat{i}}}{\Delta x^2}$ ||
 || $\alpha*_{\pm i \pm j}$ || $ 0$ ||
 
@@ -192 +192 @@
 
 
-Now in the code, 
-|| $A = A$ ||
-|| $C = \frac{\Delta t A C}{\Delta x^2}$ ||
-|| $D = \frac{\Delta t A D}{\Delta x^2}$ ||
+Now in the code, $\kappa$ is assumed constant
+|| $\mbox{A} = \kappa A$ ||
+|| $\mbox{C} = \frac{\Delta t A C}{\Delta x^2}$ ||
+|| $\mbox{D} = \frac{\Delta t A D}{\Delta x^2}$ ||
 || $\mbox{Tlambda} = T_{\pm \hat{j}}^\lambda$ ||
 || $\mbox{T} = T_{\pm \hat{j}}$ ||
 || $\mbox{T0} = T $ ||
 || $\mbox{T0lambda} = T^{\lambda}$ ||
-
-And stencil(0) has the coefficient of $T_0'$, and stencil(1:2*ndim) has the coefficients of $T_{\pm \hat{j}}'$ and source has everything else.
-
-
-
-
-Now if $\kappa$ is a constant, then we have
-
-||$\alpha_0$ || $- \displaystyle \sum_{i} \frac{2B}{\Delta x^2}$ ||
-||$\alpha_{\pm i}$ || $ \frac{B}{\Delta x^2}$ ||
-|| $\alpha*_{\pm i \pm j}$ || $ 0$ ||
+|| $\mbox{alpha} = 1 $ ||
+
+
+And stencil(0) has the coefficient of $T_0'$, and stencil(1:2*ndim) has the coefficients of $T_{\pm \hat{j}}'$ and source has everything else.  Note that the coefficients $\alpha_0$ has a term for $+\hat{i}$ and $-\hat{i}$ but the way the code is written, it loops over each neighbor and adds the contribution from $-\hat{i}$ and $+\hat{i}$ on separate iterations.
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