Improving Particle Mesh Accuracy with Analytical Solutions for Newtonian and Modified Newtonian Dynamics

Abstract

This paper aims to improve the Particle Mesh (PM) method for Newtonian Dynamics (ND) and MOdified Newtonian Dynamics (MOND). The PM method involves discretizing the mass density on a grid using Gaussian smoothing, which is used to calculate the gravitational potential using Fast Fourier Transforms (FFTs). A major downside of Gaussian smoothing is that the mass density of two particles overlaps whenever they are in close proximity, which leads to incorrect orbital dynamics. Hence, the linearity of ND is used to correct the accelerations of particles whenever this overlap in mass density occurs. These short-range corrections involve the use of analytical expressions to eliminate the inaccurate influence of the PM method, correcting the interactions between particle pairs to direct particle-particle interactions. These corrections can introduce instabilities whenever two particles are too close in proximity and the time steps are too large. This issue is mitigated by using a softening parameter in the corrections. In MOND, the linearity in the accelerations is removed, which means that these short-range corrections cannot always be used. They may only be used in the limit of high accelerations, for which MOND reduces to ND. The short-range corrections are tested for circular and elliptical two-body systems, showing a significant reduction in error of their expected orbital trajectories. To test the limitations of the PM method under non-linear accelerations in MOND where no corrections may be applied, a wide binary system is used to determine the minimum distance between particles that still yields somewhat accurate. This also demonstrates that without the use of short-range corrections large grids are required, highlighting a major inefficiency of PM methods. Lastly, the corrections are tested in both ND and MOND using the Plummer and Miyamoto-Nagai models, which represent a sphere and a disk, respectively. The Plummer model showed significant improvements in stability in both ND and MOND. However, the Miyamoto-Nagai model shows less optimistic results in MOND due to the difficulty in placing the particles at a minimum separating particles whenever the accelerations are non-lineair. Overall, careful use of these short-range corrections allows for significant improvements in the PM method for both ND and MOND. Further research into optimizing the computational performance of FFTs and short-range corrections using tree codes is still available. Additionally, a time-adaptive PM can be implemented to prevent possible instabilities due to large time steps when using the corrections.