Wavelet Fluids

This paper introduces a novel wavelet-based framework for simulating both single-phase (e.g., smoke) and two-phase (e.g., bubbly water) flows, featuring unified boundary condition handling for free surfaces and solid obstacles. In liquid simulations, conventional pressure projection methods enforce zero-pressure Dirichlet conditions at free surfaces by solving a simplified pressure Poisson equation. However, these approaches neglect air-phase incompressibility, leading to artificial bubble collapse. Stream function methods overcome this limitation by solving a density-variable vector potential Poisson equation, ensuring incompressibility in both simulated and unsimulated regions while maintaining divergence-free liquid phases independent of solver accuracy. Yet, they triple the linear system's dimensionality and exhibit poor convergence near solid boundaries. The fundamental limitation of both methods stems from their governing equations: singularities emerge as density approaches extreme values. The pressure Poisson equation becomes ill-conditioned when density nears zero (air phase), compromising air-phase incompressibility, while the vector potential equation degrades as density approaches infinity (solid phase), impeding solid-boundary convergence. To address these singularities, we first propose a novel decomposition where zero and infinite densities are well-defined. We then reformulate this decomposition as a fixed-point iteration using density-agnostic curl-free and divergence-free projections, eliminating the need for linear system solves. The error equation is derived, and a necessary and sufficient convergence condition is established. Building on this, we develop an iterative algorithm that efficiently solves the fixed-point problem through alternating wavelet-based non-orthogonal curl-free and divergence-free projections. Additionally, we investigate orthogonal curl-free projections (e.g., Fourier methods) and their complementary divergence-free counterparts, providing a comprehensive comparison between wavelet and Fourier approaches. Our method simultaneously computes pressure and stream functions, retaining the incompressibility benefits of stream function approaches while resolving their computational inefficiencies and solid-boundary convergence issues. Experiments demonstrate our framework's ability to efficiently simulate complex two-phase phenomena, such as the glugging effect during water pouring and multi-liquid-region interactions across zero-density air.