Integration Over Curves and Surfaces Defined by the Closest Point Mapping

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.

“…where u 0 is the initial solution and u n the solution at time n∆t. The surface integrals are approximated using the singular values of the Jacobian of the closest point mapping [52].…”

Section: Cahn-hilliard On a Spherementioning

confidence: 99%

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

Petras

Ling

Piret

et al. 2019

“…It can be further related to the products of the singular values of the Jacobian matrix of P Γ , which provides an alternative, and in some cases easier way, for the computation of J. See [48].…”

Section: A Weight Function δ Compactly Supported Onmentioning

confidence: 99%

“…The geometrical information about the boundary is restricted to the computation of the Jacobian J and the closest point extension of the integrand f -both of which can be approximated easily by simple finite differencing applied to the distance function d Γ (x) at grid point x i within Γ ε . Furthermore, the smoothness of the weight function δ , along with the smoothness of the integrand will allow for higher order in h approximation of I[f ] by simple Riemann sum S h Γ [f ], see for example the discussion in [48].…”

Section: A Weight Function δ Compactly Supported Onmentioning

confidence: 99%

See 1 more Smart Citation

An implicit boundary integral method for computing electric potential of macromolecules in solvent

Zhong

Ren

Tsai

2018

Self Cite

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equation that arises in mathematical models for the electrostatics of molecules in solvent. The proposed method uses an implicit boundary integral formulation to derive a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separates the molecule and the solvent. The needed implicit surface is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.Key words. Poisson-Boltzmann equation, implicit boundary integral method, level set method, fast multipole method, electrostatics, implicit solvent model.

“…See also [14] for a related RBF method that carries out a local approximation of surface differential operators to solve PDEs on folded surfaces. In addition, the extension via the closest point mapping is used in the computation of integrals over curves and surfaces [23] and in the solution of PDEs on closed, smooth surfaces using volumetric variational principles [24].…”

Section: Introductionmentioning

confidence: 99%

An RBF-FD closest point method for solving PDEs on surfaces

Petras

Ling

Ruuth

2018

Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231 (14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.