A Particle method for stationary transport equations

TL;DR

This work introduces a mesh-free Particle method to compute stationary solutions of first-order transport equations in divergence form with a source term, by treating the spatial coordinate $x_1$ as a pseudo-time along characteristics and using a kernel-based SPH-like discretization. A representation formula along characteristics is derived, and a fully discrete scheme with explicit Euler updates for particle positions and weights is analyzed, yielding an error bound of the form $|u-u_h| \le C^{(1)} h^2 + C^{(2)} \frac{ε^d}{h^d} + C^{(3)} \frac{Δs}{h}$ under regularity and a characteristic-completeness assumption. The method is validated numerically on a 2D linear transport problem and applied to a landscape evolution model, demonstrating the ability to handle wet/dry regions where traditional finite-volume methods may struggle. The approach preserves positivity and offers a mesh-free alternative for complex geometries, with potential extensions to nonlinear regimes and parallel implementations for improved efficiency.

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.

Figures (17)

• Figure 1: Domains $\Omega_\ell\subset\Omega$ and $\Phi(\Omega_\ell) = \widetilde{U}_\ell\subset U$.
• Figure 2: Characteristic curves on the domain $\widetilde{U}_\ell$ and definition of $U_\delta$.
• Figure 3: Illustration of Lemma \ref{['lemma_x_ball']}. The set $\overline{B}_d(x,\delta)$ is represented by the green circle. The set $\Phi \left([t_x,T_x] \times \overline{B}_{d-1}(\xi_x,\gamma_x) \right)$ is in grey.
• Figure 4: Illustration of particle positions in the scheme. Particles are generated along the time-discretization of the blue dotted curve (a characteristic curve). The red points are the extremal particles inside the domain, and the green one are the extremal particles outside it.
• Figure 5: Grid of the linked-list method, of size $h$. The particles in the nine green neighboring cells of the blue point $x$ are in color. The red particles are located in the support of $y \mapsto \zeta^h(x-y)$ (represented by the red circle), the green particles are inside the green cells but outside this support.

Theorems & Definitions (24)

• Theorem 1.1
• Lemma 2.1
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• Remark 3.1
• proof
• Remark 3.2
• Theorem 3.1
• Remark 3.3