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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. [ABSTRACT FROM AUTHOR]

Abstract: It is increasingly common to encounter partial differential equations (PDEs) posed on surfaces, and standard numerical methods are not available for such novel situations. Herein, we develop a simple method for the numerical solution of such equations which embeds the problem within a Cartesian analog of the original equation, posed on the entire space containing the surface. This allows the immediate use of familiar finite difference methods for the discretization and numerical solution. The particular simplicity of our approach results from using the closest point operator to extend the problem from the surface to the surrounding space. The resulting method is quite general in scope, and in particular allows for boundary conditions at surface boundaries, and immediately generalizes beyond surfaces embedded in , to objects of any dimension embedded in any . The procedure is also computationally efficient, since the computation is naturally only carried out on a grid near the surface of interest. We present the motivation and the details of the method, illustrate its numerical convergence properties for model problems and also illustrate its application to several complex model equations. [Copyright &y& Elsevier]

Abstract: We present a new method for computing the curvature-driven motion of a curve constrained to move on a given surface. It is based on the Diffusion Generated Motion algorithm, and retains both the novel simplicity of that method, as well as the natural extension to curves with junctions, to general geometric motion laws, and to higher dimensions. The result is an extremely simple algorithm for curvature-dependent motion on surfaces, wherein the only evolution operation is linear diffusion in three-dimensional Euclidean space. [Copyright &y& Elsevier]

In this paper we construct a parametrization-free embedding technique for numerically evolving reaction–diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry. We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction–diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities. [ABSTRACT FROM AUTHOR]

Abstract: Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace–Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach. [Copyright &y& Elsevier]