118 | | However with AMR is we need coarse boundary values for E at t=0, and t=dt, and t = 2dt and we would like coarse boundary values at t=2dt to match solution from coarse grid... |
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120 | | But the coarse update involves one implicit and one explicit solve. Is there a way to interpolate the fine grid ghost zones in time without storing two time derivatives? If we store dE_i+dE_e then we could do an explicit radiative update internally (E^n^ -> E*), then an implicit radiative update (E* -> E^n+1^ using the fully updated ghost zones (E^n+1^,,g,, = E^n^,,g,,+dE,,g,,) and then store the new time derivative (dE=E^n+1^-E^n^) |
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122 | | This seems like it would work for the radiative field, but what about the internal energy terms? We have an equation for e^n+1^ that is a function of E^n+1^, e* and E* but we have no way of getting E* in the ghost zones... |
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124 | | So the solution is perhaps to store the contributions from the implicit update and the non-conservative heating terms that appear in the internal energy equation. Then we can update E in the ghost zones for the implicit and the first explicit term using the time derivative - and then use the new E to update e... Then the conservative explicit RadEnergy term can be calculated after the hydro step as well as the momentum explicit term - and then this flux can be coarsened - to keep the value of E in the coarse cells consistent with the value of E in the fine cells... Since both will have been updated by the same eDot and fluxes... |
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126 | | in which case we can rewrite the equations as |
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128 | | [[latex(\frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P\color{green}{-\lambda \nabla E})]] |
129 | | [[latex(\frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{orange}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{blue}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E})]] |
130 | | [[latex(\frac{\partial E}{\partial t} \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{orange}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E} )]] |
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132 | | For static diffusion, the terms in blue with v^2^/c can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green), and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). And the terms in orange can be solved semi-implicitly (velocity terms are explicit, while radiation energy terms are implicit) |
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