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Strong-stability-preserving (SSP) time discretization methods have a nonlinear stability property that makes them particularly suitable for the integration of hyperbolic conservation laws where discontinuous behavior is present. Optimal SSP schemes have been previously found for methods of order 1, 2, and 3, where the number of stages s equals the order p. An optimal low-storage SSP scheme with s = p = 3 is also known. In this paper, we present a new class of optimal high-order SSP and low-storage SSP Runge-Kutta schemes with s > p. We find that these schemes are ultimately more efficient than the known schemes with s = p because the increase in the allowable time step more than offsets the added computational expense per step. We demonstrate these efficiencies on a set of scalar conservation laws. AMS subject classifications. 65L06, 65M20PII. S0036142901389025 1. Introduction. The method of lines is a popular semidiscretization method for the solution of time-dependent partial differential equations (PDEs). The idea behind it is to first suitably discretize the spatial variables (e.g., by finite differences, finite volumes, finite elements, or spectral methods) to yield a set of ordinary differential equations (ODEs) in time. Then, this set of ODEs can be integrated using standard time-stepping techniques such as linear multistep or Runge-Kutta methods.Standard stability analysis for the solvers of such systems generally focuses on linear stability. Indeed, such analysis is often adequate when the desired solutions are smooth. However, solutions to hyperbolic PDEs may not be smooth: shock waves or other discontinuous behavior can develop even from smooth initial data. In such cases, standard discretizations based on linear stability analysis suffer from poor performance due to the presence of spurious oscillations, overshoots, and progressive smearing. The numerical solutions obtained from these discretizations often exhibit a weak form of instability (called nonlinear instability) resulting in unphysical behavior. Accordingly, numerical methods based on a nonlinear stability requirement are very desirable. Such methods were originally referred to as total variation diminishing (TVD) [17]; see also the subsequent articles [18,6]. However, following the more recent article [7], we refer to them in this paper as strong-stability-preserving (SSP) methods.We are interested in the development, implementation, and analysis of a new class of optimal SSP Runge-Kutta (SSPRK) time-stepping schemes for the system of

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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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In the singularly perturbed limit of an asymptotically small diffusivity ratio ε 2 , the existence and stability of localized quasi-equilibrium multispot patterns is analyzed for the Brusselator reactiondiffusion model on the unit sphere. Formal asymptotic methods are used to derive a nonlinear algebraic system that characterizes quasi-equilibrium spot patterns and to formulate eigenvalue problems governing the stability of spot patterns to three types of "fast" O(1) time-scale instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. The nonlinear algebraic system and the spectral problems are then studied using simple numerical methods, with emphasis on the special case where the spots have a common amplitude. Overall, the theoretical framework provides a hybrid asymptotic-numerical characterization of the existence and stability of spot patterns that is asymptotically correct to within all logarithmic correction terms in powers of ν = −1/ log ε. From a leading-order-in-ν analysis, and with an asymptotically large inhibitor diffusivity, some rigorous results for competition and oscillatory instabilities are obtained from an analysis of a new class of nonlocal eigenvalue problem (NLEP). Theoretical results for the stability of spot patterns are confirmed with full numerical computations of the Brusselator PDE system on the sphere using the closest point method. [39] proposed that localized peaks in the concentration of a chemical substance, known as a morphogen, could be responsible for the process of morphogenesis, which describes the development of a complex organism from a single cell. By means of a linearized analysis, he showed how stable spatially inhomogeneous patterns can develop from small perturbations of spatially homogeneous initial data for a coupled system of reaction-diffusion (RD) equations. Introduction. Turing inMotivated by this initial work, a systematic and rigorous approach has been developed over the last few decades for the analysis of small amplitude patterns in RD systems. After linearizing the RD system around a spatially uniform state to identify bifurcation points for the emergence of spatially nonuniform patterns, a weakly nonlinear analysis based on a

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationIMEX extensions of linear multistep methods with general monotonicity and boundedness properties ABSTRACT For solving hyperbolic systems with stiff sources or relaxation terms, time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms. In this paper we consider implicit-explicit (IMEX) multistep methods. Suitable methods will be constructed, based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations. Numerical comparisons are made with several implicit-explicit Runge-Kutta methods.