Stability of meshfree discretizations

Characterize the stability of meshfree discretizations used by the moving least squares particle method for the Fokker–Planck equation, including whether such discretizations are stable for every possible point cloud.

Background

The paper develops a moving least squares (MLS) particle method for the S-formulation of the Fokker–Planck equation. Because the particle cloud moves, compresses, expands, and may interact with reflective boundaries, the resulting meshfree spatial discretization can encounter irregular point configurations. The authors use adaptive time-stepping based on local Hessian eigenvalues to control the explicit particle-transport update, but this does not provide a general stability theory for the meshfree discretization.

The unresolved issue is whether the meshfree method has stability guarantees for arbitrary particle clouds. Establishing such guarantees would clarify the method’s robustness beyond the numerical configurations tested in the paper and could reduce reliance on problem-dependent adaptive time-step criteria.