Spiral-Spectral Fluid Simulation

We introduce a fast, expressive method for simulating fluids over radial domains, including discs, spheres, cylinders, ellipses, spheroids, and tori. We do this by generalizing the spectral approach of Laplacian Eigenfunctions, resulting in what we call spiral-spectral fluid simulations. Starting with a set of divergence-free analytical bases for polar and spherical coordinates, we show that their singularities can be removed by introducing a set of carefully selected enrichment functions. Orthogonality is established at minimal cost, viscosity is supported analytically, and we specifically design basis functions that support scalable FFT-based reconstructions. Additionally, we present an efficient way of computing all the necessary advection tensors. Our approach applies to both three-dimensional flows as well as their surface-based, codimensional variants. We establish the completeness of our basis representation, and compare against a variety of existing solvers.

Automated summary

In this paper, a fast and expressive method for simulating fluids over radial domains is introduced, which includes discs, spheres, cylinders, ellipses, spheroids, and tori. The method, referred to as spiral-spectral fluid simulations, generalizes the spectral approach of Laplacian Eigenfunctions and includes carefully selected enrichment functions to remove singularities and establish orthogonality at minimal cost. The method also supports viscosity analytically and includes basis functions designed to support scalable FFT-based reconstructions. Additionally, an efficient way of computing necessary advection tensors is presented, and the approach applies to both three-dimensional flows and their surface-based, codimensional variants, with completeness of basis representation established and comparison made against existing solvers.