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.

翻译：我们提出了一种基于速度的蒙特卡洛流体求解器，克服了现有基于涡量方法的局限性。由于基于速度的公式在图形学中更为常见，我们的蒙特卡洛求解器可以便捷地扩展流体模拟文献中的各种技术。我们通过算子分裂求解纳维-斯托克斯方程，并针对每个子步骤设计逐点蒙特卡洛估计器，从而推导出该方法。基于最近提出的边界游走技术[Sugimoto et al. 2023]，我们将投影和扩散步骤重构为积分问题。我们将压力泊松方程源项产生的体积积分转化为更适用于实际数值计算的形式。最终形成的基于速度的公式能够正确模拟先前的基于涡量的蒙特卡洛方法[Rioux-Lavoie and Sugimoto et al. 2022]要么错误模拟、要么无法支持的场景。我们通过展示如何直接添加浮力效应、散度控制能力以及数值耗散降低方法（如平流反射和PIC/FLIP方法），证明了该方法能够轻松融合传统非蒙特卡洛方法的先进成果。