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

doi:10.1007/s10915-022-02045-w

	Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds
	Yan, Qile 1; Jiang, Shixiao W.2 ; Harlim, John 3
	2023-01
发表期刊	JOURNAL OF SCIENTIFIC COMPUTING (IF:2.8[JCR-2023],2.7[5-Year])
ISSN	0885-7474
EISSN	1573-7691
卷号	94 期号:1
DOI	10.1007/s10915-022-02045-w
摘要	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. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
关键词	Advection Diffusion Mesh generation Diffusion maps Ghost point diffusion map Local kernel Mesh-free PDE solver Meshfree Parabolic PDE on manifold Parabolic PDEs PDE solvers Time-dependent advection
URL	查看原文
收录类别	EI ; SCOPUS
语种	英语
出版者	Springer
EI入藏号	20224713159742
EI主题词	Mathematical operators
EI分类号	723.5 Computer Applications ; 921.4 Combinatorial Mathematics, Includes Graph Theory, Set Theory
原始文献类型	Journal article (JA)
引用统计	正在获取...
文献类型	期刊论文
条目标识符	https://kms.shanghaitech.edu.cn/handle/2MSLDSTB/251774
专题	数学科学研究所_PI研究组(P)_蒋诗晓组
通讯作者	Jiang, Shixiao W.
作者单位	1.Department of Mathematics, The Pennsylvania State University, University Park; PA; 16802, United States; 2.Institute of Mathematical Sciences, ShanghaiTech University, Shanghai; 201210, China; 3.Department of Mathematics, Department of Meteorology and Atmospheric Science, Institute for Computational and Data Sciences, The Pennsylvania State University, University Park; PA; 16802, United States
通讯作者单位	数学科学研究所
推荐引用方式 GB/T 7714	Yan, Qile,Jiang, Shixiao W.,Harlim, John. Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds[J]. JOURNAL OF SCIENTIFIC COMPUTING,2023,94(1).
APA	Yan, Qile,Jiang, Shixiao W.,&Harlim, John.(2023).Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds.JOURNAL OF SCIENTIFIC COMPUTING,94(1).
MLA	Yan, Qile,et al."Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds".JOURNAL OF SCIENTIFIC COMPUTING 94.1(2023).