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Diffusion (P1 ) Equation Set

$\displaystyle \alpha$$\displaystyle {\frac{{\partial \Phi}}{{\partial t}}}$ - $\displaystyle \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$D$\displaystyle \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$\displaystyle \Phi$ + $\displaystyle \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{J}$}$ + $\displaystyle \sigma$$\displaystyle \Phi$ = S

Which can be written

$\displaystyle \alpha$$\displaystyle {\frac{{\partial \Phi}}{{\partial t}}}$ + $\displaystyle \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ + $\displaystyle \sigma$$\displaystyle \Phi$ = S

$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ = - D$\displaystyle \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$\displaystyle \Phi$ + $\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{J}$}$

Where

$\displaystyle \Phi$ = Intensity  
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ = Flux  
D = Diffusion Coefficient  
$\displaystyle \alpha$ = Time Derivative Coefficient  
$\displaystyle \sigma$ = Removal Coefficient  
S = Intensity Source Term  
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{J}$}$ = Flux Source Term  


next up previous
Next: Algebraic Solution Up: Equation Sets Previous: Simplified Spherical Harmonics (SPN
Michael L. Hall