A Monte Carlo Method for Fluid Simulation

Abstract

We present a novel Monte Carlo-based fluid simulation approach capable of pointwise and stochastic estimation of fluid motion. Drawing on the Feynman-Kac representation of the vorticity transport equation, we propose a recursive Monte Carlo estimator of the Biot-Savart law and extend it with a stream function formulation that allows us to treat free-slip boundary conditions using a Walk-on-Spheres algorithm. Inspired by the Monte Carlo literature in rendering, we design and compare variance reduction schemes suited to a fluid simulation context for the first time, show its applicability to complex boundary settings, and detail a simple and practical implementation with temporal grid caching. We validate the correctness of our approach via quantitative and qualitative evaluations - across a range of settings and domain geometries - and thoroughly explore its parameters' design space. Finally, we provide an in-depth discussion of several axes of future work building on this new numerical simulation modality.