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.

翻译：本文提出了一种数值方法，用于求解包含复杂吸收体与反射体构型的无界平面区域中的扩散问题。该方法对抛物型问题应用拉普拉斯变换，得到修正的亥姆霍兹方程，并通过边界积分法进行求解。通过沿所谓塔尔博特围道变形进行逆拉普拉斯变换的数值积分，实现时域解的还原。我们演示了该方法在多种复杂几何构型中的应用，这些构型由互不相交的任意形状物体组成，其上施加均匀的狄利克雷或诺伊曼边界条件。拉普拉斯变换的使用规避了传统时间步进法的限制，使得对无界域扩散问题中存在的长时平衡过程进行积分成为可能。利用该方法，我们展示了复杂几何结构调控吸收集捕获动力学的屏蔽效应。特别地，我们通过实例说明几何结构能够引导扩散过程趋向特定吸收位点、遮蔽扩散粒子对吸收位点的感知，甚至能发现约束几何结构（如迷宫）的出口路径。