A Particle method for stationary transport equations

TL;DR

This paper introduces a particle-based numerical scheme for stationary transport equations, demonstrating convergence, error estimates, and applications to landscape evolution modeling with wet/dry area handling.

Contribution

It develops a novel particle method for stationary transport equations inspired by non-stationary methods, with proven convergence and practical applications.

Findings

Convergence of the scheme under regularity assumptions

Error estimates provided for the numerical method

Successful numerical simulations including landscape evolution

Abstract

We present and study a Particle method for the stationary solutions of a class of transport equations. This method is inspired by non-stationary Particle methods, the time variable being replaced by one spatial variable. Particles trajectories are computed using the ``time-dependent'' equations, and then the approximation is based on a quadrature method using the particle locations as quadrature points. We prove the convergence of the scheme under suitable regularity assumptions on the data and the solution, together with a ``characteristic completeness'' assumption (the characteristic curves fullfill the whole computational domain). We also provide an error estimate. The scheme is tested numerically on a two dimensional linear equation and we present a numerical study of convergence. Finally, we use this method to carry out numerical simulations of a landscape evolution model, where an erodible topography evolves under the effects of water erosion and sedimentation. The scheme is then useful to deal with wet/dry areas.