The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces

Abstract

Abstract. Many applications in the natural and applied sciences require the solutions of partial differential equations (PDEs) on surfaces or more general manifolds. The Closest Point Method is a simple and accurate embedding method for numerically approximating PDEs on rather general smooth surfaces. However, the original formulation is designed to use explicit time stepping. This may lead to a strict time-step restriction for some important PDEs such as those involving the Laplace-Beltrami operator or higher-order derivative operators. To achieve improved stability and efficiency, we introduce a new implicit Closest Point Method for surface PDEs. The method allows for large, stable time steps while retaining the principal benefits of the original method. In particular, it maintains the order of accuracy of the discretization of the underlying embedding PDE, it works on sharply defined bands without degrading the accuracy of the method, and it applies to general smooth surfaces. It also is very simple and may be applied to a rather general class of surface PDEs. Convergence studies for the in-surface heat equation and a fourth-order biharmonic problem are given to illustrate the accuracy of the method. We demonstrate the flexibility and generality of the method by treating flows involving diffusion, reaction-diffusion and fourth-order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension.

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

“…It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].…”

“…It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space.…”

Implicit boundary integral methods for the Helmholtz equation in exterior domains

“…For wider stencils, it is necessary to consider one-sided non-oscillatory approximations of the derivatives in order to minimize errors due to differencing over kinks of the distance function. The closest point mapping is used in [22] as an Eulerian method to track interfaces and in [14,20] for solving PDEs on surfaces. In the case that Γ is given as a collection of parameterized patches, one may use the fast algorithm proposed in [23] to compute the closest point mappings.…”

The Explicit–Implicit–Null method: Removing the numerical instability of PDEs

“…In [5], we presented a rough sketch of the idea, and used it to stabilize the viscous free-surface dynamics of two liquid drops during coalescence. Subsequently, similar ideas have been implemented to stabilize the motion of a surface in the diffuse interface and level-set methods [6,7,8], and for the solution of PDEs on surfaces [9].…”