Changes between Version 8 and Version 9 of u/johannjc/scratchpad5


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Timestamp:
10/01/15 13:49:51 (9 years ago)
Author:
Jonathan
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  • u/johannjc/scratchpad5

    v8 v9  
     1== Basic Equations ==
    12The anisotropic conduction equation looks like
    23
    … …  
    910$\chi_\perp = \kappa_\perp \frac{n}{B^2 T^{1/2}}$
    1011
     12== Collecting power's of T ==
     13
    1114So we can rewrite the equations as
    1215
    … …  
    1619
    1720$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
     21
     22== Einstein simplification ==
    1823
    1924or in Einstein notation
    … …  
    3136or
    3237
    33 $\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} $
     38$\partial_t T = \frac{\kappa_\parallel}{\lambda_\parallel+1}  \partial_i n b_i  b_j \partial_j T^{\lambda_\parallel+1} +   \frac{\kappa_\perp}{\left ( \lambda_\perp + 1\right )} \partial_i n^2 B^{-2} \left ( \delta_{ij} - b_i b_j \right )  \partial_j T^{\lambda_\perp +1} $
    3439
    35 Now both of these terms are of the form
     40== Implicitization ==
    3641
    37 $ A \partial_i  B_{ij} \partial_j T^{\lambda+1}$
     42Now we can rewrite the equation
     43
     44$\partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T^{\lambda+1}$
     45
     46where the $\parallel$ or $\perp$ subscript on $A$, $B_{ij}$, and $lambda$ is implied. 
     47
     48$A_\parallel = \frac{\kappa_\parallel}{\lambda_\parallel + 1}$ and $A_\perp = \frac{\kappa_\perp}{\lambda_\perp + 1}$
     49
     50and $B_{\parallel, ij} = n b_i b_j$ and $B_{\perp, ij} = n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$
    3851
    3952Now to solve this implicitly, we need to replace $T$ with $T_*$ where
    … …  
    4356Note for Backward Euler, $\phi = 1$ and for Crank Nicholson, $\phi = 1/2$
    4457
    45 $ A \partial_i  B_{ij} \partial_j T_*^{\lambda+1}$
     58$ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i  B_{ij} \partial_j T_*^{\lambda+1}$
    4659
    47 so we have
    48 
    49 $A \partial_i  B_{ij} \partial_j \left ((1-\phi) T + \phi T' \right )^{\lambda+1}$
    50 
    51 Now to solve this using a linear system, we need to linearize terms involving $T'$
    52 
    53 So we need to Taylor expand about $T$
    54 
    55 $\left ((1-\phi) T + \phi T' \right )^{\lambda+1} \approx T^{\lambda + 1} + \left ( \lambda + 1 \right ) T^{\lambda} \phi \left ( T' - T \right ) $
    56 
    57 or
    58 
    59 $\left ((1-\phi) T + \phi T' \right )^{\lambda+1} \approx \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T'$
    60 
    61 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
    62 
     60It is simpler to linearize the equation in terms of $T_*$ and then subsitute then vice-versa - though both give the same answer
    6361
    6462$T_*^{\lambda + 1} \approx T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \left ( T_{*} - T \right ) =  T^{\lambda + 1} +  \left ( \lambda + 1 \right ) T^{\lambda} \phi \left ( T' - T \right )$
    … …  
    6866So we have
    6967
    70 $A \partial_i  B_{ij} \partial_j \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ]$
     68$\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \partial_j \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ]}$
    7169
    72 $A \partial_i  B_{ij} \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right ) \partial_j T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) \partial_j T^{\lambda} T' \right ]$
     70$\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right ) \partial_j T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) \partial_j T^{\lambda} T' \right ]}$
    7371
    74 $A \partial_i  B_{ij} \left [ C_\lambda  \partial_j T^{\lambda + 1} + D_\lambda  \partial_j T^{\lambda} T' \right ]$
     72$\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i  B_{ij} \left [ C \partial_j T^{\lambda + 1} + D \partial_j T^{\lambda} T' \right ]}$
    7573
    7674where $ C= \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )$ and $ D = \phi \left ( \lambda + 1 \right )$
    … …  
    7876Now we can expand the derivatives and get
    7977
    80 $A \left ( \partial_i  B_{ij} \right ) \left ( C  \partial_j T^{\lambda + 1} + D  \partial_j T^{\lambda} T' \right ) + A B_{ij} \left ( C \partial_i \partial_j T^{\lambda + 1} + D  \partial_i \partial_j T^{\lambda} T' \right ) $
     78== Expand Derivatives ==
    8179
    82 and the whole equation is then
    83 
    84 $\partial_t T = \displaystyle \sum_{\parallel,\perp}{A \left ( \partial_i  B_{ij} \right ) \left ( C  \partial_j T^{\lambda + 1} + D  \partial_j T^{\lambda} T' \right ) + A B_{ij} \left ( C  \partial_i \partial_j T^{\lambda + 1} + D  \partial_i \partial_j T^{\lambda} T' \right )} $
    85 
    86 where $A$, $B$, $C$, $D$, and $\lambda$ are functions of $\perp,\parallel$
     80$\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \left ( \partial_i  B_{ij} \right ) \left ( C  \partial_j T^{\lambda + 1} + D  \partial_j T^{\lambda} T' \right ) + A B_{ij} \left ( C \partial_i \partial_j T^{\lambda + 1} + D  \partial_i \partial_j T^{\lambda} T' \right )} $
    8781
    8882We can also write this as
    8983
    90 $\partial_t T = \displaystyle \sum_{\parallel,\perp}{CE \partial_j T^{\lambda + 1} + DE \partial_j T^{\lambda} T' + CF  \partial_i \partial_j T^{\lambda + 1} + DF \partial_i \partial_j T^{\lambda} T'} $
     84$\partial_t T = \displaystyle \sum_{\parallel,\perp}{CE_j \partial_j T^{\lambda + 1} + DE_j \partial_j T^{\lambda} T' + CF_{ij}  \partial_i \partial_j T^{\lambda + 1} + DF_{ij} \partial_i \partial_j T^{\lambda} T'} $
    9185
    9286where
    … …  
    9690$F_{ij} = A B_{ij}$
    9791
     92
     93== Discretization ==
    9894
    9995We then need expressions for
    … …  
    110106
    111107
    112 We can also write the equation as
     108Using the above definitions, we can write the discretized equation as
    113109
    114110
    115 $T_0 -  \displaystyle \sum_{\parallel, \perp} C \left [ \alpha_0 T^{\lambda+1}_0 -  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} -  \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
    116 $T'_0 + \displaystyle \sum_{\parallel, \perp} D \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
     111$T_0 -  \displaystyle \sum_{\parallel, \perp} C \left [ \alpha_0 T^{\lambda+1}_0 -  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} -  \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
     112$T'_0 + \displaystyle \sum_{\parallel, \perp} D \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
    117113
    118114where
    … …  
    124120$\alpha_{\pm i, \pm j} = \pm \pm \frac{F_{ij}\Delta t}{4 \Delta x^2}$
    125121
     122
     123== Summary ==
     124
     125||  ||   $\parallel$   ||   $\perp$   ||
     126|| $A$ || $\frac{\kappa_\parallel}{\lambda_\parallel + 1}$ || $\frac{\kappa_\perp}{\lambda_\perp + 1}$ ||
     127|| $B$ || $n b_i b_j$ || $n^2 B^-2 b_i b_j \left ( 1 - \delta_{ij} \right )$ ||
     128|| $C$ ||  $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ ||  $\left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ ||
     129|| $D$ ||  $\phi \left ( \lambda_\parallel + 1 \right )$ ||  $\phi \left ( \lambda_\perp + 1 \right ) $||
     130
     131And then using those we can calculate
     132
     133|| $E$ || $A \left ( \partial_i  B_{ij} \right )$ ||
     134|| $F$ || $A B_{ij}$ ||
     135||$\alpha_0 $ ||$ -\frac{2F_{ij}\delta_{ij} \Delta t}{\Delta x^2}$ ||
     136||$\alpha_{\pm j}$ || $ \pm \frac{E_j \Delta t}{2 \Delta x} + \frac{F_{jj}\Delta t}{\Delta x^2}$ ||
     137||$\alpha_{\pm i, \pm j}$ || $ \pm \pm \frac{F_{ij}\Delta t}{4 \Delta x^2}$ ||
     138