Volume average of gravitational force. Double integral problem.
The background to this problem is that I have a computational code that is used to model fluid flow on a Cartesian computational grid. I have a disk of material that is in rotation about a central point and the material is gravitationally bound.
My problem is that I had been calculating the gravitational force at the center of each grid cell which will have an error as the force of gravity is inversely proportional to .
I'm trying to make my calculations more accurate by taking a volume integral of the gravitational force as the errors can be fairly large when r is fairly small.
The force due to gravity is given by , where G and M are constants.
The fact that I'm working on a cartesian grid (which I have no choice about) makes the double integral of Fgrav with x and y as the variables hard and I'm not sure how to go about this one, the limits of the integral will the the vertices of a square basically. Has anyone any suggestions on how to tackle it?
Thanks in advance for any advice.