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We develop a multi-dimensional, parallelized domain decomposition strategy (DDC) for mass-transfer particle tracking (MTPT) methods. These methods are a type of Lagrangian algorithm for simulating reactive transport and are able to be parallelized by employing large numbers of CPU cores to accelerate run times. In this work, we investigate different procedures for "tiling" the domain in two and three dimensions, (2-d and 3-d), as this type of formal DDC construction is currently limited to 1-d. An optimal tiling is prescribed based on physical problem parameters and the number of available CPU cores, as each tiling provides distinct results in both accuracy and run time. We further extend the most efficient technique to 3-d for comparison, leading to an analytical discussion of the effect of dimensionality on strategies for implementing DDC schemes. Increasing computational resources (cores) within the DDC method produces a trade-off between inter-node communication and on-node work. For an optimally subdivided diffusion problem, the 2-d parallelized algorithm achieves nearly perfect linear speedup in comparison with the serial run up to around 2700 cores, reducing a 5-hour simulation to 8 seconds, and the 3-d algorithm maintains appreciable speedup up to 1700 cores., Comment: 26 pages, 21 figures, Submitted to Journal of Computational Physics (JCP)

Traditional probabilistic methods for the simulation of advection-diffusion equations (ADEs) often overlook the entropic contribution of the discretization, e.g., the number of particles, within associated numerical methods. Many times, the gain in accuracy of a highly discretized numerical model is outweighed by its associated computational costs or the noise within the data. We address the question of how many particles are needed in a simulation to best approximate and estimate parameters in one-dimensional advective-diffusive transport. To do so, we use the well-known Akaike Information Criterion (AIC) and a recently-developed correction called the Computational Information Criterion (COMIC) to guide the model selection process. Random-walk and mass-transfer particle tracking methods are employed to solve the model equations at various levels of discretization. Numerical results demonstrate that the COMIC provides an optimal number of particles that can describe a more efficient model in terms of parameter estimation and model prediction compared to the model selected by the AIC even when the data is sparse or noisy, the sampling volume is not uniform throughout the physical domain, or the error distribution of the data is non-IID Gaussian., Comment: 24 pages

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., Comment: 26 pages, 11 figures

Geochemical systems are known to exhibit highly variable spatiotemporal behavior. This may be observed both in non-smooth concentration curves in space for a single sampling time and also in variability between samples taken from the same location at different times. However, most models that are designed to simulate these systems provide only single-solution smooth curves and fail to capture the noise and variability seen in the data. We apply a recently developed reactive particle-tracking method to a system that displays highly-complex geochemical behavior. When the method is made to most closely resemble a corresponding Eulerian method, in its unperturbed form, we see near-exact match between solutions of the two models. More importantly, we consider two approaches for perturbing the model and find that the spatially-perturbed condition is able to capture a greater degree of the variability present in the data. This method of perturbation is a task to which particle methods are uniquely suited and Eulerian models are not well-suited. Additionally, because of the nature of the algorithm, noisy spatial gradients can be highly resolved by a large number of mobile particles, and this incurs negligible computational cost, as compared to expensive chemistry calculations., Comment: 29 pages, 8 figures

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.

A significant drawback of Lagrangian (particle-tracking) reactive transport models has been their inability to properly simulate interactions between solid and liquid chemical phases, such as dissolution and precipitation reactions. This work addresses that problem by implementing a mass-transfer algorithm between mobile and immobile sets of particles that allows aqueous species of reactant that are undergoing transport to interact with stationary solid species. This mass-transfer algorithm is demonstrated to solve the diffusion equation and thus does not introduce any spurious mixing. The algorithm is capable of simulating an arbitrarily small level of diffusion, and can be combined with diffusive random walks to simulate the desired level of diffusion in a reactive transport system., Comment: 43 pages, 15 figures

Several algorithms have been used for mass transfer between particles undergoing advective and macro-dispersive random walks. The mass transfer between particles is required for general reactions on, and among, particles. The mass transfer is shown to be diffusive, and may be simulated using implicit, explicit, or mixed methods. All algorithms investigated are accurate to $\mathcal{O}(\Delta t)$. For $N$ particles, the implicit and semi-implicit methods require inverse matrix solutions and $\mathcal{O}(N^3)$ calculations. The explicit methods use forward matrix solves and require only $\mathcal{O}(N^2)$ calculations. Practically, this means that naive implementations with more than about 5,000 particles run more reliably using explicit methods, Comment: 14 pages, 3 figures