Boundary Integral Methods for Particle Diffusion in Complex Geometries: Shielding, Confinement, and Escape

TL;DR

This work introduces a boundary-integral, Laplace-transform framework for deterministic diffusion in unbounded planar domains with disjoint absorbing and reflecting bodies. By transforming to the modified Helmholtz equation and solving via a well-conditioned boundary-integral equation system, the authors compute the time-domain density and the flux through absorbing regions using an inverse Laplace transform along a Talbot contour. The method demonstrates geometry-driven shielding and routing of diffusion, including Faraday-cage-like confinement and maze-like exit selection, while delivering accurate long-time statistics without time-stepping. The results highlight the approach’s potential for complex geometries and mixed boundary conditions, with clear avenues for extensions to Robin or mixed BCs, general initial conditions, and moving geometries.

Abstract

We present a numerical method for the solution of diffusion problems in unbounded planar regions with complex geometries of absorbing and reflecting bodies. Our numerical method applies the Laplace transform to the parabolic problem, yielding a modified Helmholtz equation which is solved with a boundary integral method. Returning to the time domain is achieved by quadrature of the inverse Laplace transform by deforming along the so-called Talbot contour. We demonstrate the method for various complex geometries formed by disjoint bodies of arbitrary shape on which either uniform Dirichlet or Neumann boundary conditions are applied. The use of the Laplace transform bypasses constraints with traditional time-stepping methods and allows for integration over the long equilibration timescales present in diffusion problems in unbounded domains. Using this method, we demonstrate shielding effects where the complex geometry modulates the dynamics of capture to absorbing sets. In particular, we show examples where geometry can guide diffusion processes to particular absorbing sites, obscure absorbing sites from diffusing particles, and even find the exits of confining geometries, such as mazes.

Figures (12)

• Figure 1: A schematic of the domain configuration comprising Neumann $\Gamma_N = \cup_{k=1}^{M_N} \Gamma_{N_k}$ and Dirichlet $\Gamma_D = \cup_{k=1}^{M_D} \Gamma_{D_k}$ components and source location ${\mathbf{x}}^{\ast}\in\Omega$.
• Figure 2: The Talbot (solid black) contour \ref{['eqn:talbot']}. Integration along the Talbot contour is done with the midpoint rule, and the black marks are the resulting quadrature points with $M = 24$. The singularities of $P({\mathbf{x}},s)$ are along the negative real axis (solid red).
• Figure 3: The real and imaginary parts of a typical integrand required to compute a Talbot integral. The truncation of the integrand and the choice of the Talbot contour guarantee that the quadrature error is $\mathcal{O}\left(10^{-1.2M}\right)$, until $M$ is sufficiently large that machine precision is reached.
• Figure 4: A verification of the convergence of Talbot integration on the exactly solvable 1D heat equation \ref{['eqn:1DPeriodicDiffusion']}. The left plot shows the exact and approximate solutions, and the right plot shows the error at four different times as a function of the number of Talbot quadrature points. For $M \leq 12$, the error converges as $10^{-1.2 M}$ (black dashed curve). If the same Talbot contour were used for larger values of $M$, round-off error would begin to dominate the quadrature error. Therefore, for these larger values of $M$, the Talbot contour is slightly modified Weideman2015.
• Figure 5: A spatial convergence study of the BIE solver at four target points for increasing number of collocation points $N$. The black circle in the middle of the inset is the absorbing body. Each blue curve represents a particular $s$-value on the Talbot contour at time $t=10$. The dashed black lines indicate third-order convergence. If the modulus of the target (blue point) is too close (gray region) to the modulus of the initial particle location (red point), then the exact solution has round-off error which results in the loss of convergence for large $N$.