A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow

TL;DR

This paper introduces a new discrete model for vortex filament motion that preserves integrability and improves real-time smoke simulation algorithms in computer graphics.

Contribution

It presents a novel doubly discrete analogue of smoke ring flow that enhances numerical methods for fluid simulation.

Findings

The discretisation maintains the integrable structure of the continuous system.

Improved algorithms enable more accurate real-time smoke simulation.

The model is applicable to various physical and computational contexts.

Abstract

Modelling incompressible ideal fluids as a finite collection of vortex filaments is important in physics (super-fluidity, models for the onset of turbulence) as well as for numerical algorithms used in computer graphics for the real time simulation of smoke. Here we introduce a time-discrete evolution equation for arbitrary closed polygons in 3-space that is a discretisation of the localised induction approximation of filament motion. This discretisation shares with its continuum limit the property that it is a completely integrable system. We apply this polygon evolution to a significant improvement of the numerical algorithms used in Computer Graphics.