CSE Community Seminar | February 13, 2026

Abstract

This talk presents a structure-preserving perspective on numerical methods for ideal incompressible fluid simulation, inspired by the geometric formulation of fluid dynamics. A key focus is the preservation of coadjoint orbits of the volume-preserving diffeomorphism group—a geometric way of describing how ideal fluids evolve that is closely tied to classical circulation conservation and particle-relabeling symmetry. Within this framework, I show how widely used particle–grid methods, including Particle-in-Cell (PIC) and Fluid-Implicit-Particles (FLIP), can be augmented through a small number of principled geometric modifications and reinterpreted within a unified, geometry-driven discretization paradigm. This viewpoint clarifies which physical invariants are preserved by the resulting schemes, including energy and Casimir quantities, and provides a systematic path toward high-order accurate methods for two- and three-dimensional ideal fluid flows. The talk will also briefly touch on ongoing work examining the variational structure underlying these discretizations, including an interpretation based on non-holonomically constrained dynamics. This perspective provides a setting in which vakonomic principles can be explored alongside more traditional Lagrange–d’Alembert approaches to discretization, and helps situate the proposed methods within broader discussions of consistency and structure preservation in numerical fluid simulation.