Velocity-Based Monte Carlo Fluids

TL;DR

This paper introduces a velocity-based Monte Carlo fluid solver that improves upon vorticity-based methods, enabling more accurate and flexible fluid simulations by leveraging operator splitting and boundary techniques.

Contribution

The paper develops a novel velocity-based Monte Carlo solver for Navier-Stokes equations, enhancing simulation accuracy and compatibility with existing fluid simulation techniques.

Findings

Supports scenes previously challenging for vorticity-based methods

Easily incorporates buoyancy, divergence control, and dissipation reduction techniques

Demonstrates improved accuracy and flexibility in fluid simulation

Abstract

We present a velocity-based Monte Carlo fluid solver that overcomes the limitations of its existing vorticity-based counterpart. Because the velocity-based formulation is more commonly used in graphics, our Monte Carlo solver can be readily extended with various techniques from the fluid simulation literature. We derive our method by solving the Navier-Stokes equations via operator splitting and designing a pointwise Monte Carlo estimator for each substep. We reformulate the projection and diffusion steps as integration problems based on the recently introduced walk-on-boundary technique [Sugimoto et al. 2023]. We transform the volume integral arising from the source term of the pressure Poisson equation into a form more amenable to practical numerical evaluation. Our resulting velocity-based formulation allows for the proper simulation of scenes that the prior vorticity-based Monte Carlo method [Rioux-Lavoie and Sugimoto et al. 2022] either simulates incorrectly or cannot support. We demonstrate that our method can easily incorporate advancements drawn from conventional non-Monte Carlo methods by showing how one can straightforwardly add buoyancy effects, divergence control capabilities, and numerical dissipation reduction methods, such as advection-reflection and PIC/FLIP methods.