Above:
Top row: r = 100, r = 200, r = 1000 basis functions are used. Bottom row: r = 2000, r = 4000, r = 12600 basis functions are used. Previous approaches have only been able to achieve r ≈ 500, and would have needed 2.25 TB of memory to simulate the r = 12600 case.

Abstract

The Laplacian Eigenfunction method for fluid simulation, which we refer to as Eigenfluids, introduced an elegant new way to capture intricate fluid flows with near-zero viscosity. However, the approach does not scale well, as the memory cost grows prohibitively with the number of eigenfunctions. The method also lacks generality, because the dynamics are constrained to a closed box with Dirichlet boundaries, while open, Neumann boundaries are also needed in most practical scenarios. To address these limitations, we present a set of analytic eigenfunctions that supports uniform Neumann and Dirichlet conditions along each domain boundary, and show that by carefully applying the discrete sine and cosine transforms, the storage costs of the eigenfunctions can be made completely negligible. The resulting algorithm is both faster and more memory-efficient than previous approaches, and able to achieve lower viscosities than similar pseudo-spectral methods. We are able to surpass the scalability of the original Laplacian Eigenfunction approach by over two orders of magnitude when simulating rectangular domains. Finally, we show that the formulation allows forward scattering to be directed in a way that is not possible with any other method.

This work was supported by NSF CAREER award IIS-1253948, as well as IIS-1321168 and IIS-1619376. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We acknowledge support from the Center for Scientific Computing from the CNSI, MRL: an NSF MRSEC (DMR-1720256).