A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

JOURNAL OF COMPUTATIONAL PHYSICS

卷:381

Article

Petras, A.^1,4  Ling, L.³  Piret, C.²  Ruuth, S. J.¹ 

[1] Simon Fraser Univ, Dept Math, Burnaby, BC V5A 1S6, Canada

[2] Michigan Technol Univ, Dept Math Sci, Houghton, MI 49931 USA

[3] Hong Kong Baptist Univ, Dept Math, Kowloon Tong, Hong Kong, Peoples R China

[4] BCAM, Bilbao 48009, Basque Country, Spain

关键词: Partial differential equations on moving surfaces;    Closest point method;    Grid based particle method;    Radial basis functions finite differences (RBF-FD);    Least-squares method;   

DOI  :  10.1016/j.jcp.2018.12.031

来源: Elsevier

【 摘 要 】

The closest point method (Ruuth and Merriman (2008) [32]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao (2009) [37]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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