Changes between Version 66 and Version 67 of SelfGravityDevel

Timestamp:

10/24/19 11:31:17 (5 years ago)

Author:

Jonathan

Comment:

—

SelfGravityDevel

@@ -2 +2 @@
 
 == Physics ==
-=== Poisson Equation of Self Gravity ===
+=== Equations of Self Gravity ===
 
 The inclusion of self gravity involves an additional force to the momentum equation
@@ -8 +8 @@
 [[latex($\frac{d \rho \mathbf{v}}{dt} = \mathbf{f}_g = -\rho \nabla \phi$)]]
 
-and the energy equation now includes the gravitational self-energy on the left
+and the energy equation now includes the work done by gravity
 
 [[latex($\frac{d E}{dt} = \mathbf{v} \cdot \mathbf{f}_g = -\rho \mathbf{v} \cdot \nabla \phi$)]]
@@ -22 +22 @@
 
 
+== Making the equations conservative ==
+
 AstroBEAR uses hypre's linear solver to solve Poisson's equation for the potential.  However, there are two methods for including the source terms for momentum and energy.  The non-conservative approach simply includes the terms during a source update, while the conservative approach recasts the RHS of the momentum and energy equations as total divergences - so they can be differenced conservatively.
 
 Substituting the Poisson equation into the momentum equation we have after some manipulations
 
-
-[[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi \right) + \rho_0 \phi \right ) \right ]$)]]
+[[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \mathbf{I} \left ( \nabla \phi \cdot \nabla \phi \right) \right ) + \rho_0 \phi  \mathbf{I} \right ]$)]]
 
 where we have the momentum flux tensor 
 
-[[latex($\mathbf{T} =  \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right )$)]]
-
-
+[[latex($\mathbf{T} =  \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2}\mathbf{I} \left ( \nabla \phi \cdot \nabla \phi \right) \right) + \rho_0 \phi \mathbf{I}$)]]
 
 
@@ -75 +74 @@
 
 [[latex($\mathbf{F_g} = \phi \rho \mathbf{v} + \frac{1}{8 \pi G} \left (\phi \nabla \dot{\phi} - \dot{\phi}\nabla \phi \right) $)]]
+
+== Solving for $\dot{\phi}$ ==
+
+
+Now to solve for $\dot{\phi}$ we can solve directly 
+
+$\nabla^2 \dot{\phi} = 4 \pi G \dot{\rho} = 4 \pi G \left ( \frac{\rho^{n+1} - \rho^n}{\Delta t} \right )$ 
+
+or we can simply wait until we have solved for the new potential based on the updated density 
+
+$\nabla^2 \phi^{n+1} = 4 \pi G \rho^{n+1}$ 
+
+and then use 
+
+$\dot{\phi} = \frac{\phi^{n+1} - \phi^n}{\Delta t}$ 
+
+to apply the energy correction.
+
+== Currently what is done in AstroBEAR ==
+
+AstroBEAR's momentum flux is first calculated using
+
+[[latex($\mathbf{T^n} =  \frac{1}{4 \pi G} \left ( \nabla \phi^n \nabla \phi^n - \frac{1}{2} \mathbf{I} \left ( \nabla \phi^n \cdot \nabla \phi^n \right ) + \rho_0 \phi^n \mathbf{I} \right )$)]]
+
+Where $\phi^{n}$ is used and everything is calculated at cell faces.  
+
+Then the energy is updated using 
+
+[[latex($E^{n+1} = E^n - \rho \mathbf{v} \cdot \nabla \phi$)]]
+
+where each component in the sum $\rho \mathbf{v} \cdot \nabla \phi$ is averaged at the left and right face for each dimension.
+
+Then following the second elliptic solve, the momentum is updated again by calculating the difference between the original momentum flux tensor $\mathbf{T}$ and the time centered one $\mathbf{T}^{n+1/2} = \mathbf{T}^n + \frac{1}{2} \left ( \mathbf{T}^{n+1} - \mathbf{T}^n \right)$
+
+And a corrective momentum flux tensor is calculated
+
+[[latex($\mathbf{T'} = \frac{1}{2} \left (T^{n+1} - T^n \right )$)]]
+
+and applied.
+
+