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

Abstract

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961) 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, J. Comput. Phys.228 (8): 2993-3024, [2009]) 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.

Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

“…In this paper, we solve the GS system on surfaces by proposing an implicit formulation in the time direction. Moreover, the RBF-FD [35,36] is considered to discretize the time-independent problem in the spatial direction.…”

“…The radial basis function-finite difference (RBF-FD) method is a hybrid and advanced procedure in numerical analysis constructed by combining the beneficent characteristics of the radial basis function (RBF) and easy implementation of finite difference [35,36]. The main factor behind the RBF-FD method's extension is to reduce the computational cost of global methods.…”

Generalized finite difference method for anomalous diffusion on surfaces

“…It is noted that this method requires minimizing eqn (9), after which we could obtain a linear system AD b u = , (11) where…”

“…However, in recent years, with increasing attractiveness of partial differential equations (PDEs) defined on surfaces or manifolds [10][11][12], there are only a few reports of fractional order equations defined on surfaces. Surface PDEs or surface operators are a type of PDEs or operators defined in tangent space, which are different in Euclidean space.…”

A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems

“…A review of finite element methods for surface PDEs can be found in [14]. Recently, Petras et al [15], [16], and Petras and Ruuth et al [19], [20] developed a systematic way to solve surface PDEs by using RBF-FD combined with a grid particle method. These methods reduce the computational workload tremendously, but still need a grid in the neighbourhood The associate editor coordinating the review of this manuscript and approving it for publication was Kathiravan Srinivasan .…”