A Scalable Fast Multipole Method Poisson Solver for the RAMSES code: II. Adaptive Mesh Refinement and Adaptive Time Stepping

We present an extended implementation of a scalable, O(N) Poisson solver based on the fast multipole method (FMM), fully compatible with adaptive mesh refinement (AMR) and adaptive time stepping (ATS) within the RAMSES framework. Building on the unigrid algorithm in Lee & Teyssier (2026), we introduce several novel elements, including the use of multiple FMM trees, one for each AMR level, a merged FMM tree for all active levels to optimize neighbor searches, and the introduction of the concept of "nearest-field" to enforce force symmetry across refinement levels. Across a broad set of test problems, we find excellent, percent-level agreement with our reference traditional multigrid (MG) solver. We show, however, that FMM exhibits better momentum conservation properties across coarse-fine AMR interfaces. Finally, despite the overhead introduced by the spatio-temporal adaptivity, FMM shows better scalability than MG across various AMR configurations, with the largest gains obtained for the largest configurations.