Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds

• Published in: Journal of scientific computing 2023, Vol.94 (1), p.5, Article 5
• Main Authors: Yan, Qile, Jiang, Shixiao W., Harlim, John
• Format: Article
• Language: English
• Subjects: Advection; Advection-diffusion equation; Algorithms; Approximation; Boundary conditions; Computational Mathematics and Numerical Analysis; Computer graphics; Convergence; Differential equations; Manifolds (mathematics); Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Operators (mathematics); Partial differential equations; Theoretical; Time dependence; Topology
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-02045-w

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold.