We present recursive algorithm to simulate fast arrival events, where the first among many diffusing particles reaches a target, Classical approaches to simulate such events rely on full trajectory generation of all particles, leading to prohibitive computational costs in the large particle number regime. Here we present simulation framework for efficiently generating order statistics of arrival times by exploiting asymptotic first-passage distributions.We start with the case of instantaneous (delta-function) emission, the algorithm simulate the first $k$ arrivals without tracking particle trajectories. We present also the algorithm to general time-dependent emission profiles. This an accelerated simulation scheme bypasses naive Brownian dynamics, with direct applications to spatial reaction networks, rare event detection, and diffusion-controlled activation.