A boundary-aware deterministic Lagrangian framework for transport

We present a boundary-aware deterministic Lagrangian framework for passive scalar transport, suited to accurately resolve concentration fields in confined geometries, which is of particular interest for the study of mixing-limited processes in porous media at the pore scale. Our formulation describes transport using time-independent random coefficients, which allows for the deterministic reconstruction of one-point statistics without Monte Carlo sampling or numerical diffusion. To incorporate the influence of impermeable irregular boundaries, we solve a hyperbolic Liouville master equation in Lagrangian form with the method of characteristics, adding specular reflections to the characteristic lines at the boundaries, which define deterministic trajectories of what we call support particles. These particles carry their corresponding value of the joint probability density function governed by the Liouville equation that, marginalized over the coefficients, gives the normalized concentration field. Spectral convergence with the number of support particles is possible thanks to their deterministic nature. We show how this approach can be used to resolve scalar mixing near boundaries of arbitrary shape, and provide smooth, stable results on arbitrary grids. This framework opens new avenues to develop accurate mesh-free solvers of advection-diffusion processes, offering a flexible and computationally efficient alternative to classical Lagrangian and Eulerian approaches.