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.

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Level set methods have been used in a great number of applications in R 2 and R 3 and it is natural to consider extending some of these methods to problems defined on surfaces embedded in R 3 or higher dimensions. In this paper we consider the treatment of level set equations on surfaces via a recent technique for solving partial differential equations (PDEs) on surfaces, the Closest Point Method [RM06]. Our main modification is to introduce a Weighted Essentially Non-Oscillatory (WENO) interpolation step into the Closest Point Method. This, in combination with standard WENO for Hamilton-Jacobi equations, gives high-order results (up to fifth-order) on a variety of smooth test problems including passive transport, normal flow and redistancing. The algorithms we propose are straightforward modifications of standard codes, are carried out in the embedding space in a well-defined band around the surface and retain the robustness of the level set method with respect to the self-intersection of interfaces. Numerous examples are provided to illustrate the flexibility of the method with respect to geometry.

We investigate the strong stability preserving (SSP) property of two-step Runge-Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge-Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations. * 4700

Abstract. In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multi-core architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on Integral Deferred Correction (IDC), which was itself motivated by Spectral Deferred Correction by Dutt, Greengard and Rokhlin (BIT-2000).The method presented here is a revised formulation of explicit IDC, dubbed Revisionist IDC, which can achieve p th -order accuracy in "wall-clock time" comparable to a single forward Euler simulation on problems where the time to evaluate the right-hand side of a system of differential equations is greater than latency costs of inter-processor communication, such as in the case of the N -body problem. The key idea is to re-write the defect correction framework so that, after initial startup costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and M = p − 1 correctors in parallel on an interval which has K steps, where K p. We prove that given an r th -order Runge-Kutta method in both the prediction and M correction loops of RIDC, then the method is order r × (M + 1).The parallelization in Revisionist IDC uses a small number of cores (the number of processors used is limited by the order one wants to achieve). Using a four-core CPU, it is natural to think about fourth-order RIDC built with forward Euler, or eighth-order RIDC constructed with secondorder Runge-Kutta. Numerical tests on an N -body simulation show that RIDC methods can be significantly faster than popular Runge-Kutta methods such as the classical fourth-order RungeKutta scheme.In a PDE setting, one can imagine coupling RIDC time integrators with parallel spatial evaluators, thereby increasing the parallelization. The ideas behind RIDC extend to implicit and semiimplicit IDC and have high potential in this area.

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. or, more generally, the elliptic operatorThe Laplace-Beltrami eigenvalue problem has played a prominent role in recent years in data analysis. For example, in [1], eigenvalues of the Laplace-Beltrami operator were used to extract "fingerprints" which characterize surfaces and solid objects. In [2, 3], Laplace-Beltrami eigenvalues and eigenfunctions were used for dimensionality reduction and data representation. Other application areas include smoothing of surfaces [4] and the segmentation and registration of shape [5].Analytical solutions to the Laplace-Beltrami eigenvalue problem are rarely available, so it is crucial to be able to numerically approximate them in an accurate and efficient manner. Partial differential equations on surfaces, including eigenvalue problems, have traditionally been approximated using either (a) discretizations based on a parameterization of the surface [6], (b) finite element discretizations on a triangulation of the surface [7], or (c) embedding techniques which solve some embedding PDE in a small region near the surface [8] (see also the related works [9,10,11,12,13,14,15]).Parameterization methods (a) are often effective for simple surfaces [6], but for more complicated geometries have the deficiency of introducing distortions and singularities into the method through the parameterization [16]. Approaches based on the finite element method can be deceptively difficult to implement; as described in [7], "even though this method seems to be very simple, it is quite tricky to implement". Embedding methods (c) have gained a considerable following because they permit PDEs on surfaces to be solving using standard finite differences.This paper proposes a simple and effective embedding method for the Laplace-Beltrami eigenvalue problem based on the Closest Point Method. The Closest Point Method is a recent embedding method that has been used to compute the numerical solution to a variety of partial differential equations [17,18,19,20], including in-surface heat flow, reaction-diffusion equations, and higher-order motions involving biharmonic and "surface diffusion" terms. Unlike traditional embedding methods, which are built around level set representatives of the surface, the Closest Point Method i...

The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization of this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs.Comment: 22 pages, 3 figures, 4 table

The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulksurface coupling are provided for a variety of complex point cloud surfaces.closest point method | embedding method | Laplace-Beltrami P artial differential equations (PDEs) are widely used to describe continuum processes such as diffusion, chemical reactions, fluid flow, or electrodynamics. In standard 3D settings, these take a familiar PDE form, such as a reaction-diffusion equation:and the ways to numerically solve such equations are welldeveloped. The basic approach can be quite simple, such as laying down a uniform Cartesian grid of points, fx i;j;k : 1 ≤ i; j; k ≤ Ng, and using simple, familiar approximations of the differential terms on this grid, such as:where u i;j;k ≈ uðx i;j;k Þ and h is the uniform grid spacing. As a result, it is very easy to implement methods for the numerical solution of such equations to study the phenomena of interest. To achieve efficiency for large-scale computations, more advanced methods are required, such as implicit discretization and solvers for systems of equations. These solvers, while internally complex, have been implemented in standard, well-validated numerical routines and are accessible through numerical subroutine libraries. Important physical processes also arise in complex geometrical settings, such as on complicated surfaces in three dimensions. The abstract form of differential operators on surfaces remains the same as in 3D, however, when explicitly expressed in coordinates, the formulas for the operators and the corresponding discretized expressions are relatively complicated and have received much less attention. Moreover, in practical settings, surfaces are often defined simply as a set of points-a point cloud-sampled from the underlying surface. Because the connectivity of the points is not provided, this adds further complexity to methods that need to reconstruct the geometric properties of the surface, such as the metric distance.Here we present a method for solving reaction-diffusion equations on a point cloud that represents the underlying surface-or any other geometric object-in a way that reduces the problem to working with entirely standard classical 3D discretizations and solver libraries. Our approach is fundamentally different from oth...