Coding with the transverse intersection algebra

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.