Adaptive Local Kernels for Lagrangian Modeling of Reactive Transport
Jueves 08 de Marzo a las 12:15 h
Departamento de Ingeniería Civil y Ambienta, Modulo D2-Aula CIHS, Planta Baja
In recent years, a number of Lagrangian approaches [e.g. 1,2,3,4,5] have been proposed for the numerical solution of nonlinear reactive transport problems. In these approaches, reactions are driven by the interaction between numerical particles that represent either a volume of fluid or a mass of solute. This interaction is a function of the relative position of particles and can be derived mathematically by equipping each particle with a kernel function, sometimes referred to as the smoothing function. In some of these approaches, the choice of the kernel shape and size does not follow objective criteria, but is instead rather heuristic. On the other hand, some authors have proposed that the smoothing function should be derived from the dispersive process [e.g. 2,6].
Recently, Fernàndez-Garcia and Sànchez-Vila  proposed that the kernel should represent the probability density distribution of the actual particle location, and should be globally optimized by minimizing the Mean Integrated Squared Error (MISE) of the density estimation. By using the existing algorithms for the determination of a time-dependent globally optimal Kernel Density Estimator (KDE), it is possible to compute probabilities (or rates) of reaction of numerical particles to model simple bimolecular reactions  or kinetic reactions of any complexity involving two reactants .
Most research on KDE focuses on 1D distributions of limited complexity. However, solute concentration distributions in heterogeneous media exhibit complex features and variations in space and time. For this reason, the globally optimal kernel is often far from being locally optimal, in particular, in those parts of the domain with relative small particle densities, and also in those with particularly abrupt concentration changes (such as mixing fronts). A novel approach is presented to define an adaptive locally optimal kernel function in 1-, 2- or 3-D that is not only time-dependent but also space-dependent. The presented local approaches are compared against the existing global ones, and also against the simple binning method, in the context of a Random Walk Particle Tracking (RWPT) model of reactive transport through a heterogeneous porous medium. The results show that the use of a locally adaptive kernel has a considerably positive impact on the accuracy of concentration estimations and chemical reaction simulations.
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