  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.
