Self Gravity in astrobear

The governing equation is Poisson's equation for gravity, which for a given mass distribution can be solved for the gravitational potential. In astrobear, we use the potential to solve for the gravitational forces in the fluid. The equation we use for the resultant force is either in conservative or non-conservative form (want to read more into the numerical methods here and link to these pages).

Laplace Operator

In cartesian coordinates, the Laplacian contains only simple derivatives, i.e. does not contain any functions of position as it does in cylindrical or spherical coordinates:

in Cartesian coordinates (x,y,z):

\nabla^2 = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} + \frac{\partial^2f}{\partial z^2}

in Cylindrical coordinates (\rho, \phi, z):

\nabla^2 = \frac{1}{\rho} \frac{\partial}{\partial \rho}(\rho \frac{\partial f}{\partial \rho}) + \frac{1}{\rho ^2}\frac{\partial^2 f }{\partial \phi^2} + \frac{\partial^2f}{\partial z^2}

in Spherical (r, \theta, \phi):

\nabla^2 = \frac{1}{r ^2} \frac{\partial}{\partial r}(r^2 \frac{\partial f}{\partial r}) + \frac{1}{r ^2 \sin \theta}\frac{\partial}{\partial \theta}(\sin \theta \frac{\partial f}{\partial \theta}) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \phi ^2}

and in 2.5 d:

Poisson Equation takes on a simple discretization form in Cartesian coordinates since the Laplacian contains only simple derivatives.