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.

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.

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