Changes between Version 145 and Version 146 of FluxLimitedDiffusion

Timestamp:

03/31/13 17:05:54 (13 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -537 +537 @@
 - & \left ( \psi \left (  \alpha^n_{i-1/2} - \zeta^n_{i-1/2} v^n_{x,i-1/2} + \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n+1}_{i-1}  \\
 = & \left ( 1 - \bar{\psi} \left ( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} - \zeta^n_{i+1/2} v^n_{x,i+1/2} +  \zeta^n_{i-1/2} v^n_{x,i-1/2} + \frac{\epsilon - \xi}{1-\psi \phi} \right ) \right ) E^{n}_i  \\
-+ & \left ( \bar{\psi} \left (  alpha^n_{i+1/2} + \zeta^n_{i+1/2} v^n_{x,i+1/2} - \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n}_{i+1}  \\
-+ & \left ( \bar{\psi} \left (  alpha^n_{i-1/2} - \zeta^n_{i-1/2} v^n_{x,i-1/2} + \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n}_{i-1}  \\
++ & \left ( \bar{\psi} \left (  \alpha^n_{i+1/2} + \zeta^n_{i+1/2} v^n_{x,i+1/2} - \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n}_{i+1}  \\
++ & \left ( \bar{\psi} \left (  \alpha^n_{i-1/2} - \zeta^n_{i-1/2} v^n_{x,i-1/2} + \frac{\omega v^n_x}{1-\psi \phi} \right ) \right ) E^{n}_{i-1}  \\
 - & \frac{\theta}{1-\psi \phi}   \\
 \end{eqnarray}
 }}}
 
-
-Now since the second equation has no spatial dependence, we can solve it for 
-   [[latex(\color{purple}{e^{n+1}_i = \frac{1}{ 1 +\psi \phi^n_i}\left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( \psi \omega_i v_{x,i} \right ) E^{n+1}_{i+1} - \left ( \psi \omega_i v_{x,i} \right ) E^{n+1}_{i-1} + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n + \bar{\psi} \omega_i v_{x,i}  \left ( E^{n}_{i+1}- E^{n}_{i-1} \right ) \right \}} )]]   
-
-and plug the result into the first equation to get a matrix equation involving only one variable.
-
-   [[latex(\color{purple}{\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \frac{\epsilon^n_i}{ 1 +\psi \phi^n_i}\right ) \right ] E^{n+1}_i - \left ( \psi \left ( \alpha^n_{i+1/2}- \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \right ) E^{n+1}_{i+1} - \left ( \psi \left ( \alpha^n_{i-1/2} + \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \right ) E^{n+1}_{i-1} =\left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2}  + \frac{\epsilon^n_i }{ 1 +\psi \phi^n_i} \right ) \right ] E^n_i + \left ( \bar{\psi} \left ( \alpha^n_{i+1/2} - \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \right ) E^{n}_{i+1} + \left ( \bar{\psi} \left ( \alpha^n_{i-1/2} + \frac{\omega_i v_{x,i}}{1+\psi \phi^n_i} \right ) \right ) E^{n}_{i-1} + \frac{ \phi^n_i}{ 1 +\psi \phi^n_i}  e^n_i+ \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n})]]   
-
-
 === 2D etc... ===
 
-For 2D or 3D we have more connections to add to the matrix elements but it is very straight forward...  There will be additional alpha terms for each dimension - and the velocity components (v,,x,,) will change, but everything else stays the same.
+For 2D or 3D we have more connections to add to the matrix elements but it is very straight forward...  There will be additional terms for each dimension - and the velocity components (v,,x,,) will change, but everything else stays the same.
 
 
 === Initial solution vector ===
-For the initial solution vector, we can just use Edot from the parent update (or last time step if we are on the coarse grid) to guess E, and then we can solve for the new e given our guess at the new E using the same time stepping (Backward Euler, Crank Nicholson, etc...).
-
+For the initial solution vector, we can just use Edot from the parent update (or last time step if we are on the coarse grid) to guess E
 [[latex(E^{n+1}_i = E^{n}_i+\dot{E}^{n}_i \Delta t)]]
-[[latex(e^{n+1}_i = \frac{1}{1+\psi \phi^n_i} \left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n \right \} )]]
 
 == Coarse-Fine Boundaries ==
@@ -570 +559 @@
 
 == Physical Boundary Conditions ==
+For each boundary type we need to specify [[latex(E^n_g, E^{n+1}_g, \kappa_g, \mbox { and } v_g)]] in terms of other quantities (ie including E^n+1^_i).  The v,,g,, and kappa,,g,, terms should come from the hydro boundary conditions so we just need an equation for E^n^,,g,,
+
 
 === Open (Free streaming) boundaries ===
-We would like the radiation to leave at the free streaming limit.  So 
+We would like the radiation to leave at the free streaming limit.  But we could have to modify the opacity in the ghost zone to keep the radiation energy positive
+
 [[latex(\frac{c \lambda}{\kappa_{0R}} \nabla E = \mathbf{F} = cE\mathbf{n} = \frac{c \lambda_g}{\kappa_g}\frac{ \left ( E-E_g \right ) }{\Delta x})]]
 
-Clearly if we set [[latex(E_g = 0)]] and [[latex(\lambda_g =\kappa_g \Delta x)]] we should get the correct flux.
-
-This corresponds to an
+Clearly if we set [[latex(E_g = 0)]] and [[latex(\kappa_g << \frac{1}{\Delta x} )]] we should get [[latex(\lambda = \kappa_g \Delta x)]] and a free streaming flux of [[latex(c E \mathbf{n})]] 
+
+which also corresponds to an
 [[latex(\alpha = c \frac{\Delta t}{\Delta x})]]
 
-So we would just modify [[latex(\alpha)]] and zero out the matrix coefficient to the ghost zone
-
-
-
 === Constant Slope Boundary ===
-Here we want the flux to be constant so energy does not pile up near the boundary.  If we cancel all derivative terms on both sides of the cell, this will effectively match the incoming flux with the outgoing flux.  This can also be done by setting [[latex(\alpha_g = \alpha_i = 0)]]
+[[latex(E_g = 2 E_i - E_{i+1})]]
 
 === Periodic Boundary ===
@@ -595 +583 @@
 
 === Reflecting/ZeroSlope Boundary ===
-Reflecting boundary should be fairly straightforward.  This an be achieved by setting [[latex(\alpha_g = 0)]] which zeros out any flux - and has the same effect as setting [[latex(E^{*}_g=E^{*}_i)]] or [[latex(E^{n+1}_g=E^{n+1}_i \mbox{ and } E^{n}_g=E^{n}_i)]]
+
+[[latex(E_g = E_i)]]
 
 === Constant radiative flux ===
-To have a constant radiative flux we must zero out terms involving the gradient and just add [[latex(F_0 \frac{\Delta t}{\Delta x})]] in the source vector
-
-=== Summary ===
-
-
-|| Numerical value  ||  Boundary                  ||  [[latex(E^{n+1}_{i+1})]]        ||  [[latex(E^{n+1}_i)]]                         ||   [[latex(E^{n+1}_{i-1})]]                       ||  [[latex(S)]] 
-||  0               ||  RAD_FREE_STREAMING        ||  [[latex(\color{red}{0})]]                    ||  [[latex(\color{red}{\psi c \frac{\Delta t}{\Delta x}})]]  ||  [[latex(-\psi\alpha_{i-1/2})]]  ||  [[latex(\color{red}{-\bar{\psi}c \frac{\Delta t}{\Delta x} E^n_i} + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]    || 
-||  1               ||  RAD_EXTRAPOLATE_FLUX     ||  [[latex(\color{red}{0})]]                    ||  [[latex(\color{red}{0})]]                                 ||  [[latex(\color{red}{0})]]  ||  [[latex(\color{red}{0})]]    ||
-||  2               ||  INTERNAL/PERIODIC         ||  [[latex(-\psi\alpha_{i+1/2})]]  ||  [[latex(\psi\alpha_{i+1/2})]]                ||  [[latex(-\psi\alpha_{i-1/2})]]  ||  [[latex(\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}-E^n_i \right ) + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]  ||
-||  3               ||  RAD_REFLECTING            ||  [[latex(\color{red}{0})]]                    ||  [[latex(\color{red}{0})]]                                 ||  [[latex(-\psi\alpha_{i-1/2})]]  ||  [[latex( \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]    ||
-||  4               ||  RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY  ||  [[latex(\color{red}{0})]]       ||  [[latex(\psi\alpha_{i+1/2})]]                ||  [[latex(-\psi\alpha_{i-1/2})]]  ||   [[latex(\bar{\psi}\alpha_{i+1/2} \left ( E^n_{i+1}-E^n_i \right ) \color{red}{- \psi \alpha_{i+1/2} E^{n+1}_{i+1}} + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]   ||
-||  5               ||  RAD_USERDEFINED_FLUX      ||  [[latex(\color{red}{0})]]                    ||  [[latex(\color{red}{0})]]                                 ||   [[latex(-\psi\alpha_{i-1/2})]]  ||    [[latex(\color{red}{F_0 \frac{\Delta t}{\Delta x}} + \bar{\psi}\alpha_{i-1/2} \left ( E^n_{i-1}-E^n_i \right ))]]   ||
-
-
-
-And for the semi implicit term
-
-|| Numerical value  ||  Boundary                  ||  [[latex(E^{n+1}_{i+1})]]        ||  [[latex(E^{n+1}_{i})]]        ||   [[latex(E^{n+1}_{i-1})]]     ||   [[latex(S)]] 
-||  0               ||  RAD_FREE_STREAMING        ||    [[latex(0)]]                    ||  [[latex(\frac{\psi \omega_i v_{x,i}}{1+\psi \phi^n_i}-\frac{\psi \epsilon^n_i \left ( 2 - \frac{\kappa_{0R,i}}{\kappa_{0P,i}} \right ) \frac{v_{x,i}}{c}}{1+\psi \phi^n_i})]]  ||  [[latex(-\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  || [[latex(-\frac{\bar{\psi}\omega_i v_{x,i}}{1+\psi \phi^n_i} \left ( E^n_{i}-E^n_{i-1} \right ) + \frac{\bar{\psi} \epsilon^n_i \left ( 2 - \frac{\kappa_{0R,i}}{\kappa_{0P,i}} \right ) \frac{v_{x,i}}{c}}{1+\psi \phi^n_i} E^n_i)]]  ||
-||  1               ||  RAD_EXTRAPOLATE_FLUX*     ||  [[latex(0)]]  ||  [[latex(2\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  ||  [[latex(-2\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  ||  [[latex(-2\frac{\bar{\psi}\omega_i v_{x,i}}{1+\psi \phi^n_i} \left ( E^n_{i}-E^n_{i-1} \right ))]]  ||
-||  2               ||  INTERNAL/PERIODIC         ||  [[latex(\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  ||  [[latex(0)]]  ||  [[latex(-\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  ||  [[latex(-\frac{\bar{\psi}\omega_i v_{x,i}}{1+\psi \phi^n_i} \left ( E^n_{i+1}-E^n_{i-1} \right ))]]  ||
-||  3               ||  RAD_REFLECTING            ||  [[latex(0)]]                    ||  [[latex(\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  ||  [[latex(-\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  || [[latex(-\frac{\bar{\psi}\omega_i v_{x,i}}{1+\psi \phi^n_i} \left ( E^n_{i}-E^n_{i-1} \right ))]]  ||
-||  4               ||  RAD_USERDEFINED_BOUNDARY/AMR_BOUNDARY  ||  [[latex(0)]]       ||   [[latex(0)]]   ||   [[latex(-\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  ||  [[latex(-\frac{\bar{\psi}\omega_i v_{x,i}}{1+\psi \phi^n_i} \left ( E^n_{i+1}-E^n_{i-1} \right )  - \frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i} E^{n+1}_{i+1})]]   ||
-||  5               ||  RAD_USERDEFINED_FLUX      ||  [[latex(0)]]                    ||  [[latex(\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  ||  [[latex(-\frac{\psi\omega_i v_{x,i}}{1+\psi \phi^n_i})]]  || [[latex(-\frac{\bar{\psi}\omega_i v_{x,i}}{1+\psi \phi^n_i} \left ( E^n_{i}-E^n_{i-1} \right ))]]  ||
-
-!* We can make the left boundary look like the right boundary by reflecting the domain in x.  Then we just swap every ,,i+1,, <-> ,,i-1,, and v,,x,, -> -v,,x,,
-
-[[CollapsibleEnd()]]
+
+[[latex(E_g = E_i - F_0 \frac{\kappa_g \Delta x}{c \lambda_g}
+
+[[CollapsibleEnd()]]