n-body simulation parallel

**Parallel N-Body Simulation: Direct O(N²) and Hierarchical Methods — GPU acceleration for astrophysics and molecular dynamics** N-body simulation computes pairwise gravitational or electrostatic forces among N particles. Direct all-pairs computation requires O(N²) force evaluations, making GPU acceleration essential for systems exceeding thousands of particles. Hierarchical methods like Barnes-Hut reduce complexity to O(N log N) via spatial tree approximation. **GPU Direct N-Body Implementation** CUDA kernels for direct N-body implement O(N²) all-pairs force computation with particles partitioned into tiles. Each thread block computes forces on a tile of destination particles by loading source particles iteratively into shared memory, achieving hide latency through shared memory reuse. A single source particle interacts with multiple destination particles via register staging. Tiling improves bandwidth utilization: for N=4096 particles, naive global memory access requires ~67 billion transactions versus ~520 million with shared memory tiling (128x improvement). Timestep integration (position update) follows force computation. **Barnes-Hut Tree Acceleration** Barnes-Hut algorithms construct octree spatial hierarchies at each timestep, grouping distant particles into center-of-mass approximations. Traversal from root enables selective force computation (far particles use approximate forces, near particles compute exact pairwise forces). Tree construction, cost estimation, and traversal parallelize across particles with irregular workloads—some particles traverse deep trees while others terminate early at coarse levels. **Particle Mesh Ewald Method** PME decomposes long-range forces into short-range pairwise (computed directly) and long-range mesh-based terms (computed via FFT). This O(N log N) approach dominates molecular dynamics: short-range forces parallelize trivially, long-range forces leverage parallel FFT. Reciprocal-space spline interpolation maps particles to mesh, forward FFT, reciprocal-space multiplication, inverse FFT, and force grid interpolation back to particles. **Multi-Particle Domain Decomposition** Large distributed simulations employ spatial domain decomposition: each process owns particles in spatial regions, communicates force updates at boundaries, and load-balances through domain repartitioning as particles migrate between regions.