Alternating Direction Implicit Method for Two-Dimensional Fokker-Planck Equation of Dense Spherical Stellar Systems

TL;DR

This paper introduces an ADI numerical scheme for the 2D Fokker-Planck equation in dense stellar systems, improving stability and reducing computation time compared to traditional methods.

Contribution

The paper develops a new ADI integration scheme for the 2D FP equation, enhancing numerical stability and efficiency in modeling stellar system evolution.

Findings

Reduces computation time by approximately 50%.

Resolves numerical instability issues in 2D FP simulations.

Effective for extreme initial conditions and tidal shock effects.

Abstract

The Fokker-Planck (FP) model is one of the commonly used methods for studies of the dynamical evolution of dense spherical stellar systems such as globular clusters and galactic nuclei. The FP model is numerically stable in most cases, but we find that it encounters numerical difficulties rather often when the effects of tidal shocks are included in two-dimensional (energy and angular momentum space) version of the FP model or when the initial condition is extreme (e.g., a very large cluster mass and a small cluster radius). To avoid such a problem, we have developed a new integration scheme for a two-dimensional FP equation by adopting an Alternating Direction Implicit (ADI) method given in the Douglas-Rachford split form. We find that our ADI method reduces the computing time by a factor of ~2 compared to the fully implicit method, and resolves problems of numerical instability.