Changes between Version 1 and Version 2 of u/johannjc/scratchpad4

Timestamp:

09/30/15 16:16:44 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad4

@@ -28 +28 @@
 Alternatively, we can rewrite the diffusion equation
 
-$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{\kappa_\parallel}{\lambda+1} \left ( \hat{b} \cdot \nabla T^{\lambda+1} \right )\right ]$
+$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{\kappa_\parallel}{\lambda+1} \left ( \hat{b} \cdot \nabla T_*^{\lambda+1} \right )\right ]$
+
+
 
 and then perform a single Taylor expansion
@@ -38 +40 @@
 $\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( -\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}T_{*} \right )  $
 
+$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( -\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}T_{*} \right )  $
 
-$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \left [ \left ( \partial_i n b_i b_j \right )   \left ( -\lambda  \partial j T^{\lambda+1} + \left ( \lambda + 1 \right )  \partial j T^{\lambda}T_{*} \right )  +  n b_i b_j \left ( -\lambda \partial_i \partial j T^{\lambda+1} + \left ( \lambda + 1 \right ) \partial_i \partial j T^{\lambda}T_{*} \right ) \right ]$
+Let's also take a moment to write $T_* = \phi T + \psi T'$  where $T'$ is the new temperature, and $\phi + \psi = 1$.  Backward Euler would have $\phi=0$ and $\psi=1$ where Crank-Nicholson would have $\phi=\psi=1/2$
+
+$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( -\lambda T^{\lambda+1} + \left ( \lambda + 1 \right ) T^{\lambda}\left ( \phi T + \psi T' \right ) \right )  $
 
 
+$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \partial_i  n b_i b_j \partial j \left ( \left ( \phi - \psi \lambda \right ) T^{\lambda+1} + \psi \left ( \lambda + 1 \right ) T^{\lambda} T' \right )   $
 
 
+$\partial_t T = \frac{\kappa_\parallel}{\lambda+1} \left [ \left ( \partial_i n b_i b_j \right )  \partial j \left ( \left ( \phi - \psi \lambda \right ) T^{\lambda+1} + \psi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ) +  n b_i b_j \left ( \left ( \phi - \psi \lambda \right ) \partial_i \partial_j T^{\lambda+1} + \psi \left ( \lambda + 1 \right ) \partial_i \partial_j T^{\lambda} T' \right ) \right ]$
+
+
+Now if we write the equation as 
+
+$\partial_t T = B_i \partial_j T^{\lambda+1} + C_j \partial_j T^\lambda T' + D_{ij} \partial_i\partial_j T^{\lambda+1} + E_{ij} \partial_i \partial_j T^\lambda T'$
+
+we get expressions for
+
+$B_j =  \frac{\kappa_\parallel \left ( \phi-\psi\lambda \right )}{\lambda+1} \partial_i n b_i b_j $
+
+$C_j =   \kappa_\parallel \psi \partial_i n b_i b_j $
+
+$D_{ij}=\frac{\kappa_\parallel \left ( \phi-\psi\lambda \right )}{\lambda+1} n b_i b_j $
+
+$E_{ij} =   \kappa_\parallel \psi  n b_i b_j $
+
+We then need expressions for
+
+$ \partial_t T = \frac{T'_0 - T_0}{\Delta t}$
+$ \partial_j T^{\lambda+1} = \frac{T^{\lambda+1}_{\hat{j}} - T^{\lambda+1}_{-\hat{j}}}{2 \Delta x}$
+
+$ \partial_j T^{\lambda}T' = \frac{T^{\lambda}_{\hat{j}}T'_{\hat{j}} - T^{\lambda}_{-\hat{j}}T'_{-\hat{j}}}{2 \Delta x}$
+
+$\partial_i\partial_j T^{\lambda+1} = \frac{T^{\lambda+1}_{\hat{i}+\hat{j}} - T^{\lambda+1}_{\hat{i}-\hat{j}}-T^{\lambda+1}_{-\hat{i}+\hat{j}} + T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2}\left(1-\delta_{ij}\right ) +  \frac{T^{\lambda+1}_{\hat{i}} - 2T^{\lambda+1}_{0} + T^{\lambda+1}_{-\hat{i}}}{\Delta x^2}\delta_{ij}$
+
+$\partial_i\partial_j T^{\lambda}T' = \frac{T^{\lambda}_{\hat{i}+\hat{j}}T'_{\hat{i}+\hat{j}} - T^{\lambda}_{\hat{i}-\hat{j}}T'_{\hat{i}-\hat{j}}-T^{\lambda}_{-\hat{i}+\hat{j}}T'_{-\hat{i}+\hat{j}} + T^{\lambda}_{-\hat{i}-\hat{j}}T'_{-\hat{i}-\hat{j}}}{4 \Delta x^2}\left(1-\delta_{ij}\right ) +  \frac{T^{\lambda}_{\hat{i}}T'_{\hat{i}} - 2T^{\lambda}_{0}T'_{0} + T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2}\delta_{ij}$
+
+We can also write the equation as 
+
+$\alpha_0 T'_0 + \displaystyle \sum_{\pm i} \alpha_{\pm i} T'_{\pm \hat{i}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T'_{\pm \hat{i} \pm \hat{j}} = \beta$ 
+
+and then work out the coefficients for the matrix equation
+