  Recent advances in random-walk particle-tracking have enabled direct
simulation of mixing and reactions on particles by allowing the particles to
interact with each other using a multi-point mass transfer scheme. The mass
transfer scheme allows separation of mixing and spreading processes, among
other advantages, but it is computationally expensive because its speed depends
on the number of interacting particle pairs. This note explores methods for
relieving the computational bottleneck caused by the mass transfer step, and we
use these algorithms to develop a new parallel, interacting particle model. The
new model is a combination of a sparse search algorithm and a novel
domain-decomposition scheme, both of which offer significant speedup relative
to the reference case--even when they are executed serially. We combine the
strengths of these methods to create a parallel particle scheme that is highly
accurate and efficient with run times that scale as $1 / P$ for a fixed number
of particles, where $P$ is the number of computational cores being used. The
new parallel model is a significant advance because it enables efficient
simulation of large particle ensembles that are needed for environmental
simulations, and also because it can naturally pair with parallel geochemical
solvers to create a practical Lagrangian tool for simulating mixing and
reactions in complex chemical systems.
