Solute transport in fluids can be simulated with Lagrangian methods such as Random Walk Particle Tracking(RWPT). These methods are virtually free of numerical dispersion and instabilities, which makes them particularlywell-suited to advection-dominated problems, and hence an attractive alternative to Eulerian approaches. However,incorporating nonlinear reactions in random walk models is not a straightforward task, since solute concentrationsare not readily available. A simple appr...
Solute transport in fluids can be simulated with Lagrangian methods such as Random Walk Particle Tracking(RWPT). These methods are virtually free of numerical dispersion and instabilities, which makes them particularlywell-suited to advection-dominated problems, and hence an attractive alternative to Eulerian approaches. However,incorporating nonlinear reactions in random walk models is not a straightforward task, since solute concentrationsare not readily available. A simple approach for estimating the concentrations from particle positions would be toperform a simple binning, but this generates an estimation error. On the other hand, Kernel methods have beendemonstrated to necessitate much lower particle numbers to reach the same degree of accuracy. A Gaussian kernelrepresenting the dispersion Green’s function in a time-step would virtually produce no over-smoothing bias, but,in many situations, it may require the use of prohibitive particle numbers for the solution to converge, especiallyfor non-linear reaction systems. Thus, a wider, adaptive kernel that locally modifies its size and shape so as tooptimally balance the noise and bias of the estimation can be a practical alternative. That is to assume that aparticle is only a subsample of a much larger population and thus it represents a “cluster” of particles distributedover some support volume. Recent improvements in this area have led us to the development of the Space-TimeAdaptive Reaction Supports (STARS) methodology, which can be used to generate density estimations efficiently,in 1, 2 or 3 dimensions, while accounting for the effect of boundary conditions. This methodology performs a pilotbinning density estimation, and then smooths it using the local optimal kernel smoothers, whose size depends onx through the local particle distribution features, and evolves with time. With the link between particle positionsand solute concentrations, we can implement all kinds of nonlinear reactions in random-walk models, includingequilibrium reaction systems. Finally, we introduce the publicly available Matlab class STARS.m, which can beappended to a particle-based code to generate density estimations using the presented methodology.