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90
Explicit Timestepping for stiff ODEs
 SIAM J. Sci. Comp
, 2003
"... Abstract. We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stabili ..."
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Cited by 21 (6 self)
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Abstract. We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvement is large. We illustrate the technique on a number of wellknown stiff test problems.
Constraintdefined manifolds: a legacy code approach to lowdimensional computation
 J. Sci. Comp
"... If the dynamics of an evolutionary differential equation system possess a lowdimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximat ..."
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Cited by 19 (8 self)
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If the dynamics of an evolutionary differential equation system possess a lowdimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of “slow ” variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the ChafeeInfante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.
Stability analysis and applications of the exponential time differencing schemes
 J. Comput. Math
"... Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday Exponential time differencing schemes are time integration methods that can be efficiently combined with spatial spectral approximations to provide very high resolution to the smooth solutions of some linear and nonlinear part ..."
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Cited by 16 (3 self)
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Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday Exponential time differencing schemes are time integration methods that can be efficiently combined with spatial spectral approximations to provide very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. We study in this paper the stability properties of some exponential time differencing schemes. We also present their application to the numerical solution of the scalar AllenCahn equation in two and three dimensional spaces. Mathematics subject classification: Key words: 1.
A MULTISCALE METHOD FOR HIGHLY OSCILLATORY ORDINARY DIFFERENTIAL EQUATIONS WITH RESONANCE IN MEMORY OF GERMUND DAHLQUIST
"... ABSTRACT. A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly ..."
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Cited by 16 (2 self)
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ABSTRACT. A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to ɛ perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam. 1.
Patch dynamics with buffers for homogenization problems
 J. Computational Physics
, 2006
"... An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such timedependent multiscale problems, an “equationfree ” framework has been proposed, of which patch dynamics is an essential component. Patch dyna ..."
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Cited by 14 (4 self)
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An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such timedependent multiscale problems, an “equationfree ” framework has been proposed, of which patch dynamics is an essential component. Patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover only a fraction of the spacetime domain. To reduce the effect of the artificially introduced box boundaries, we use buffer regions to “shield ” the boundary artefacts from the interior of the domain for short time intervals. We analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity, and propose a simple heuristic to determine a sufficient buffer size. The algorithm performance is illustrated through a set of numerical examples, which include a nonlinear reactiondiffusion equation and the Kuramoto–Sivashinsky equation. 1 1
Decentralised autonomic computing: Analysing selforganising emergent behaviour using advanced numerical methods
 in Proceedings of the Second International Conference on Autonomic Computing, (Los Alamitos
, 2005
"... When designing decentralised autonomic computing systems, a fundamental engineering issue is to assess systemwide behaviour. Such decentralised systems are characterised by the lack of global control, typically consist of autonomous cooperating entities, and often rely on selforganised emergent ..."
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Cited by 12 (6 self)
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When designing decentralised autonomic computing systems, a fundamental engineering issue is to assess systemwide behaviour. Such decentralised systems are characterised by the lack of global control, typically consist of autonomous cooperating entities, and often rely on selforganised emergent behaviour to achieve the requirements. A wellfounded and practically feasible approach to study overall system behaviour is a prerequisite for successful deployment. On one hand, formal proofs of correct behaviour and even predictions of the exact systemwide behaviour are practically infeasible due to the complex, dynamic, and often nondeterministic nature of selforganising emergent systems. On the other hand, simple simulations give no convincing arguments for guaranteeing systemwide properties. We describe an alternative approach that allows to analyse and assess trends in systemwide behaviour, based on socalled “equationfree ” macroscopic analysis. This technique yields more reliable results about the systemwide behaviour, compared to mere observation of simulation results, at an affordable computational cost. Numerical algorithms act at the systemwide level and steer the simulations. This allows to limit the amount of simulations considerably. We illustrate the approach by studying a particular systemwide property of a decentralised control system for Automated Guided Vehicles and we outline a road map towards a general methodology for studying decentralised autonomic computing systems. 1.
The moment map: Nonlinear dynamics of density evolution via a few moments
 SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
, 2005
"... We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale ..."
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Cited by 11 (0 self)
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We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath.
On HMMlike integrators and projective integration methods for systems with multiple time scales
 Comm. Math. Sci
, 2007
"... Abstract. HMMlike multiscale integrators and projective integration methods and are two different types of multiscale integrators which have been introduced to simulate efficiently systems with widely disparate time scales. The original philosophies of these methods, reviewed here, were quite diffe ..."
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Cited by 9 (1 self)
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Abstract. HMMlike multiscale integrators and projective integration methods and are two different types of multiscale integrators which have been introduced to simulate efficiently systems with widely disparate time scales. The original philosophies of these methods, reviewed here, were quite different. Recently, however, projective integration methods seem to have evolved in a way that make them increasingly similar to HMMintegrators and quite different from what they were originally. Nevertheless, the strategy of extrapolation which was at the core of the original projective integration methods has its value and should be extended rather than abandoned. An attempt in this direction is made here and it is shown how the strategy of extrapolation can be generalized to stochastic dynamical systems with multiple time scales, in a way reminiscent of Chorin’s artificial compressibility method and the CarParrinello method used in molecular dynamics. The result is a seamless integration scheme, i.e. one that does not require knowing explicitly what the slow and fast variables are. Key words. Multiscale integrators; HMM; projective integration methods; stiff ODEs; averaging theorems. 1.
A COMPUTATIONAL TOOL FOR THE REDUCTION OF NONLINEAR ODE SYSTEMS POSSESSING MULTIPLE SCALES
, 2005
"... Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that aids the ..."
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Cited by 8 (3 self)
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Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that aids the identification of structure in a class of systems of nonlinear ordinary differential equations (ODEs) that are commonly applied in dynamical models of physical processes. This “dominant scale” technique prioritizes consideration of the influence that distinguished “inputs ” to an ODE have on its dynamics. On this basis a sequence of reduced models is derived, where each model is valid for a duration that is determined selfconsistently as the system’s state variables evolve. The characteristic time scales of all sufficiently dominant variables are also taken into account to further reduce the model. The result is a hybrid dynamical system of reduced differentialalgebraic models that are switched at discrete event times. The technique does not rely on explicit small parameters in the ODEs and automatically detects changing scale separation both in time and in “dominance strength ” (a quantity we derive to measure an input’s influence). Reduced regimes describing the full system have quantified domains of validity in time and with respect to variation in state variables. This enables the qualitative analysis of the system near known orbits (e.g., to study bifurcations) without sole reliance on numerical shooting methods. These methods have been incorporated into a new software tool named Dssrt, which we demonstrate on a limit cycle of a synaptically driven Hodgkin–Huxley neuron model.