Changes between Version 65 and Version 66 of SelfGravityDevel

Timestamp:

10/22/19 12:31:08 (6 years ago)

Author:

Jonathan

Comment:

—

SelfGravityDevel

@@ -24 +24 @@
 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
-
-[[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right ) \right ]$)]]
-
+Substituting the Poisson equation into the momentum equation we have after some manipulations
+
+
+[[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi \right) + \rho_0 \phi \right ) \right ]$)]]
+
+where we have the momentum flux tensor 
+
+[[latex($\mathbf{T} =  \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right )$)]]