  The problem of a spatially discontinuous diffusion coefficient
($D(\boldsymbol x)$) is one that may be encountered in hydrogeologic systems
due to natural geological features or as a consequence of numerical
discretization of flow properties. To date, mass-transfer particle-tracking
(MTPT) methods, a family of Lagrangian methods in which diffusion is jointly
simulated by random walk and diffusive mass transfers, have been unable to
solve this problem. This manuscript presents a new mass-transfer (MT) algorithm
that enables MTPT methods to accurately solve the problem of discontinuous
$D(\boldsymbol x)$. To achieve this, we derive a semi-analytical solution to
the discontinuous $D(\boldsymbol x)$ problem by employing a predictor-corrector
approach, and we use this semi-analytical solution as the weighting function in
a reformulated MT algorithm. This semi-analytical solution is generalized for
cases with multiple 1D interfaces as well as for 2D cases, including a $2
\times 2$ tiling of 4 subdomains that corresponds to a numerically-generated
diffusion field. The solutions generated by this new mass-transfer algorithm
closely agree with an analytical 1D solution or, in more complicated cases,
trusted numerical results, demonstrating the success of our proposed approach.
