It\^o tracers: continuous-trajectory Lagrangian particles for Eulerian hydrodynamics

TL;DR

The paper introduces the Itô tracer, a continuous-time Lagrangian particle method for hydrodynamics that improves upon traditional tracers by better representing advection and diffusion, validated through tests showing enhanced statistical accuracy.

Contribution

Itô tracers are a novel continuous-trajectory Lagrangian particle method that matches gas advection and diffusion, offering advantages over Monte-Carlo tracers and enabling advanced computational techniques.

Findings

Itô tracers reproduce or improve statistical measures compared to MC tracers.

They enhance correlation between tracers and gas in turbulence simulations.

They reduce the width of the log-density ratio PDF by nearly 50%.

Abstract

Lagrangian tracer particles have long been used to track the history of individual gas parcels in hydrodynamical codes. Particles advected by the cell-centered velocity carry no representation of underlying numerical diffusion, and thus exhibit systematic bias. The Monte-Carlo (MC) tracer resolves this with discrete probabilistic cell-to-cell, flux-based jumps, at the cost of trajectories that are discontinuous in time. We introduce the It\^o tracer, a continuous-time Lagrangian particle with moments matched to the advection, diffusion, and dispersion of the gas. A subgrid-scale variant (SGS-It\^o) replaces the numerical diffusion with a Smagorinsky--Lilly turbulent diffusivity, illustrating that the form of the diffusion matters less than its magnitude. We validate these methods with a 1D square-pulse advection test and 3D decaying turbulence at $\sigma_{\rm rms} = 15\,c_{\rm s}$. We compare the different tracer particle methods using several statistical tests. It\^o tracers largely reproduce or improve upon MC tracers statistics across column-density maps, joint density histograms, log-density-ratio PDFs, and density power spectra. In the turbulence test, It\^o tracers improve the correlation between tracers and gas over the MC tracers by >3\%, and reduce the width of the log-density ratio PDF by nearly 50\%. Relative to classical tracers, these improvements are $\gtrsim$30\% and 230\%, respectively. Because It\^o tracers follow a stochastic differential equation, the method maps onto other continuous-trajectory Lagrangian processes (e.g. dust grains, charged particles, cosmic rays), admits variance-reduction techniques, higher-order integrators, and GPU-friendly implementations -- all of which are unavailable to discrete-jump schemes.