Time Derivatives of Potentials...

So I think I figured out why the particles generated little bubbles when they are initially formed and it has to do with a missing factor of 2 in the force :). At first I thought it was the way potentials were stored/updated. and what follows explains how to better do this, but the potential of a line charge is so the force is proportional to not . And here is the explanation of how potentials are stored/updated…

When there are no sink particles, the gas potential , and it's time derivative are used to interpolate the solution for at coarse fine boundaries. When a sink particle is formed the gas potential should respond to the missing material. The elliptic solver will make this adjustment provided that the boundary value for is correct. However this is problematic at the coarse fine boundaries since the value of at the coarse fine boundary was prolongated from the parent grid pre-accretion. If instead of storing and using at the coarse-fine boundaries, we use then this time derivative of the combined potential should be unchanged under the conversion of gas material to sink particles - however (and this is the tricky part), this is only true if both potentials use the same constant of integration… The elliptic solver normalizes the potential so that the sum is zero. However, the potential of a sink cylinder goes like ln(r) and cannot be conveniently normalized. One could calculate the integral of the potential over the entire box and subtract off the mean value analytically (or numerically), however if a multipole expansion is used instead then this becomes unecessary. A simpler normalization could be used for the potential. where a is chosen so that the integral over the box of is zero. For a circle of radius , the constant . If we compare the potential of a particle at the center of the grid with an overdense cell at the center of the grid we see that there is fairly good agreement if we use

Here are two lineouts through the center of the grid with a particle and an overdense cell

And here is the same but with a softened particle potential

And here is the result if the particle is placed at [.125, .125]

And here is the full 2D image of the potentials

Not the periodic nature of the gas potential causes the turnover. Doing this with image particles is not generally feasible.

And here is the result of a simulation with the correct particle force. Click here​ for a movie.

Attachments (6)

Comments

No comments.