Coding with the transverse intersection algebra
Ofir Aharoni, Daniel An, Alice Kwon, Ruth Lawrence, Dennis Sullivan
TL;DR
This work develops a finite-dimensional discretization of the continuum Euler dynamics by constructing a fluid algebra from the cubical transverse intersection algebra on a cubic lattice with odd period $N$. It defines a discrete fluid algebra on $V=\{*\\partial x: x\in FC_2\}$ and derives an explicit Euler equation $\dot X=\\pi(i(X\\wedge *\\partial X))$, with a Poisson-solver map $\\pi$ that enforces the prescribed intersection structure. The paper provides complete combinatorial machinery, invariant quantities $(X,X)$ and $(X,DX)$, and explicit lattice formulas for the discrete operators, alongside a concrete roadmap for numerical implementation. Preliminary simulations with RK4 exhibit energy blow-up, suggesting stiffness and motivating implicit schemes or alternative discretizations such as TIAs for improved stability. Overall, the approach offers a principled, geometry-driven discretization of 3D incompressible Euler dynamics with directly computable invariants and explicit implementation details.
Abstract
The concept of a fluid algebra was introduced by Sullivan over a decade ago as an algebraic construct which contains everything necessary in order to write down a form of the Euler equation, as an ODE whose solutions have invariant quantities which can be identified as energy and enthalpy. The natural (infinite-dimensional) fluid algebra on co-exact 1-forms on a three-dimensional closed oriented Riemannian manifold leads to an Euler equation which is equivalent to the classical Euler equation which describes non-viscous fluid flow. In this paper, the recently introduced transverse intersection algebra associated to a cubic lattice of An-Lawrence-Sullivan is used to construct a finite-dimensional fluid algebra on a cubic lattice (with odd periods). The corresponding Euler equation is an ODE which it is proposed is a `good' discretisation of the continuum Euler equation. This paper contains all the explicit details necessary to implement numerically the corresponding Euler equation. Such an implementation has been carried out by our team and results are pending.