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.

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