Implicit energy update¶
FargoCPT code (old):
\(\alpha = 1 + 2 H \frac{\sigma_R}{c} 4 (\frac{m_\mu (\gamma - 1)}{R \Sigma})^4 e^3\)
\(\frac{\partial e}{\partial t} = \frac{-p\nabla u + Q^+ - Q^-}{\alpha}\)
What is \(\alpha\)¶
Equation to solve: \(\frac{\partial (e + E_{rad})}{\partial t} = -p\nabla u + Q^+ - Q^- - \nabla F\)¶
\(\frac{\partial e}{\partial t} = - p \nabla u + Q^+\)
\(\frac{\partial E_{rad}}{\partial t} = - Q^- - \nabla F\)
\(E_{rad} = 2 H \frac{\sigma_R}{c} T^4\)
\(T = \frac{m_\mu(\gamma - 1)}{R\Sigma} e\)
\(E_{rad} = 2 H \frac{\sigma_R}{c} (\frac{m_\mu (\gamma - 1)}{R \Sigma})^4 e^4\)
\(\frac{\partial E_{rad}}{\partial t} = 2 H \frac{\sigma_R}{c} 4 (\frac{m_\mu (\gamma - 1)}{R \Sigma})^4 e^3 \frac{\partial e}{\partial t}\)
\(\frac{\partial (e + E_{rad})}{\partial t} = -p\nabla u + Q^+ - Q^- - \nabla F\)
\(\frac{\partial e}{\partial t} + 2 H \frac{\sigma_R}{c} 4 (\frac{m_\mu (\gamma - 1)}{R \Sigma})^4 e^3 \frac{\partial e}{\partial t}= -p\nabla u + Q^+ - Q^- - \nabla F\)
\(\frac{\partial e}{\partial t}(1 + 2 H \frac{\sigma_R}{c} 4 (\frac{m_\mu (\gamma - 1)}{R \Sigma})^4 e^3)= -p\nabla u + Q^+ - Q^- - \nabla F\)
\(\alpha = 1 + 2 H \frac{\sigma_R}{c} 4 (\frac{m_\mu (\gamma - 1)}{R \Sigma})^4 e^3\)
\(\frac{\partial e}{\partial t} = \frac{-p\nabla u + Q^+ - Q^- - \nabla F} {\alpha}\)
Questions¶
Is this operation splitting still correct?¶
\(\frac{\partial e}{\partial t} = \frac{-p\nabla u + Q^+ - Q^-}{\alpha}\)
\(\frac{\partial e}{\partial t} = -p\nabla u\)
\(\frac{\partial e}{\partial t} = \frac{Q^+ - Q^-}{\alpha}\)
What is the actually emitted energy?¶
\(Q^-\) or \(\frac{Q^-}{\alpha}\)