Lanczos Advection: Near-Orthogonal Semi-Lagrangian Reconstruction for Fluid Simulation

Producing vivid fluid details, such as the sharp spatial structure of interfaces and vortices, is essential for high-quality fluid simulations in computer graphics. The advection solver is a vital component of fluid simulation, and the semi-Lagrangian method is widely adopted as a central building block for many advanced advection methods thanks to its unconditional stability. Despite these notable improvements, the numerical dissipation and energy loss inherent in conventional linear interpolation remain considerable drawbacks, compromising visual fidelity. This paper proposes a novel extension of the core semi-Lagrangian advection algorithm to significantly alleviate numerical dissipation and energy loss. Our extension uses a dimension-splitting weighted sum with a windowed sinc (also known as Lanczos) kernel, leveraging the properties of the sinc function to construct a bounded, near-orthogonal semi-Lagrangian reconstruction. Our experiments, comparing various advection methods and their discrete operators, demonstrate that our extension significantly improves the orthogonality of the operator, energy-retaining behavior and enables richer visual details.