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Advanced numerical methods for simulating advection-diffusion equations using parallelized Lagrangian particle tracking algorithms
Schauer, Lucas
Schauer, Lucas
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2024
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Lagrangian particle tracking schemes allow a wide range of flow and transport processes to be simulated accurately, but a major challenge is numerically implementing the inter-particle interactions in an efficient manner. Such methods were originally derived from a probabilistic or first-principles perspective and have previously lacked a more rigorous derivation arising directly from the underlying advection-diffusion-reaction equation (ADRE). Herein, we provide a rigorous derivation of the MTPT method as a Lagrangian approximation of solutions to the ADRE. Numerically, this research describes the development of multi-dimensional, parallelized domain decomposition (DDC) strategies for mass-transfer particle tracking (MTPT) methods in which particles exchange mass dynamically. We show that this method can be efficiently parallelized by employing large numbers of CPU cores to accelerate run times. We first validate the approach and our theoretical predictions by focusing our efforts on a well known benchmark problem with pure diffusion, where analytical solutions in any number of dimensions are well established. We are then able to extend these studies to more complex systems where analytic solutions may not exist. In particular, the MTPT methods we use can simulate systems with highly heterogeneous velocity fields and non-constant hydrodynamic dispersion. This combination of capabilities allows us to validate and expand on the existing lamella theory that describes concentration evolution via kinematic stretching, which is currently limited to a constant-dispersion assumption. Given the capability of our particle tracking methods to simultaneously simulate mixing and spreading via a velocity-dependent, hydrodynamic dispersion tensor, this research explores the existing assumptions of lamella theory and exhibits expanded high performance computing (HPC) techniques for load balancing and computationally efficiency.
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