Changes between Version 1 and Version 2 of u/johannjc/scratchpad7


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Timestamp:
10/12/15 13:09:18 (9 years ago)
Author:
Jonathan
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  • u/johannjc/scratchpad7

    v1 v2  
    8181Now we can expand the derivatives and get
    8282
    83 == Expand Derivatives ==
    84 
    85 $\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 )} $
    86 
    87 We can also write this as
    88 
    89 $\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'} $
    90 
    91 where
    92 
    93 $E_j = A \left ( \partial_i  B_{ij} \right )$
    94 
    95 $F_{ij} = A B_{ij}$
    96 
    97 
    9883== Discretization ==
    9984
    … …  
    10287$ \partial_t T = \frac{ T'_0 - T_0}{\Delta t}$
    10388
    104 $ \partial_j T^{\lambda+1} = \frac{T^{\lambda+1}_{\hat{j}} - T^{\lambda+1}_{-\hat{j}}}{2 \Delta x}$
    105 
    106 $ \partial_j T^{\lambda}T' = \frac{T^{\lambda}_{\hat{j}}T'_{\hat{j}} - T^{\lambda}_{-\hat{j}}T'_{-\hat{j}}}{2 \Delta x}$
    107 
    108 $\partial_i\partial_j T^{\lambda+1} = \frac{T^{\lambda+1}_{\hat{i}+\hat{j}} - T^{\lambda+1}_{\hat{i}-\hat{j}}-T^{\lambda+1}_{-\hat{i}+\hat{j}} + T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2}\left(1-\delta_{ij}\right ) +  \frac{T^{\lambda+1}_{\hat{i}} - 2T^{\lambda+1}_{0} + T^{\lambda+1}_{-\hat{i}}}{\Delta x^2}\delta_{ij}$
    109 
    110 $\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}$
    111 
     89$ \partial_i B_{ij} \partial_j T^{\lambda+1} = \frac{B^{ij}_{\hat{i}} T^{\lambda+1}_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda+1}_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda+1}_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2}$
     90
     91$ \partial_i B_{ij} \partial_j T^{\lambda}T' = \frac{B^{ij}_{\hat{i}} T^{\lambda}_{\hat{i}+\hat{j}}T'_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda}_{\hat{i}-\hat{j}}T'_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda}_{-\hat{i}+\hat{j}}T'_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda}_{-\hat{i}-\hat{j}}T'_{-\hat{i}-\hat{j}}}{4 \Delta x^2}$
    11292
    11393Using the above definitions, we can write the discretized equation as