# Numerical Equivalence Between SPH and Probabilistic Mass Transfer Methods for Lagrangian Simulation of Dispersion

## TL;DR

This paper demonstrates that mass transfer particle tracking methods can be mathematically unified with smoothed particle hydrodynamics by choosing an appropriate Gaussian kernel, leading to exact solutions for dispersion in certain conditions.

## Contribution

It unifies MTPT and SPH methodologies, showing their equivalence with a specific kernel choice, and validates this with numerical simulations matching analytical solutions.

## Key findings

- Kernel bandwidth of size $\ell=\sqrt{2D\Delta t}$ yields exact dispersion displacement.
- Mass transfer operation becomes exact under dense particle distribution with the optimal kernel.
- Numerical results agree with analytical solutions, confirming the theoretical unification.

## Abstract

Several Lagrangian methodologies have been proposed in recent years to simulate advection-dispersion of solutes in fluids as a mass exchange between numerical particles carrying the fluid. In this paper, we unify these methodologies, showing that mass transfer particle tracking (MTPT) algorithms can be framed within the context of smoothed particle hydrodynamics (SPH), provided the choice of a Gaussian smoothing kernel whose bandwidth depends on the dispersion and the time discretization. Numerical simulations are performed for a simple dispersion problem, and they are compared to an analytical solution. Based on the results, we advocate for the use of a kernel bandwidth of the size of the characteristic dispersion length $\ell=\sqrt{2D\Delta t}$, at least given a "dense enough" distribution of particles, for in this case the mass transfer operation is not just an approximation, but in fact the exact solution, of the solute's displacement by dispersion in a time step.