In this paper, we present an abstract framework which describes algebraically the derivation of order conditions independently of the nature of differential equations considered or the type of integrators used to solve them. Our structure includes a Hopf algebra of functions, whose properties are used to answer several questions of prime interest in numerical analysis. In particular, we show that, under some mild assumptions, there exist integrators of arbitrarily high orders for arbitrary (modified) vector fields.

In long-time numerical integration of Hamiltonian systems,and especially in molecular dynamics simulation,it is important that the energy is well conserved. For symplecticintegrators applied with sufficiently small step size, thisis guaranteed by the existence of a modifiedHamiltonian that is exactly conserved up to exponentially smallterms. This article is concerned with the simplifiedTakahashi-Imada method, which is a modificationof the Störmer-Verlet method that is as easy to implement buthas improved accuracy. This integrator is symmetric andvolume-preserving, but no longer symplectic. We study itslong-time energy conservation and give theoreticalarguments, supported by numerical experiments, whichshow the possibility of a drift in the energy (linear or like a random walk).With respect to energy conservation, this article provides empiricaland theoretical data concerning the importance of using a symplecticintegrator.

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris,SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732],we developeda class of iterative algorithmswithin the contextof equation-free methodsto approximatelow-dimensional,attracting,slow manifoldsin systemsof differential equationswith multiple time scales.For user-specified valuesof a finite numberof the observables,the mth memberof the classof algorithms( $m = 0, 1, \ldots$ )finds iterativelyan approximationof the appropriate zeroof the (m+1)st time derivativeof the remaining variablesanduses this rootto approximate the locationof the pointon the slow manifoldcorresponding to these valuesof the observables.This articleis the firstof two articlesin whichthe accuracy and convergenceof the iterative algorithmsare analyzed.Here,we work directlywith fast-slow systems,in which there isan explicit small parameter, ε,measuring the separationof time scales.We show that,for each $m = 0, 1, \ldots$ ,the fixed pointof the iterative algorithmapproximates the slow manifoldup to and includingterms of ${\mathcal O}(\varepsilon^m)$ .Moreover,for each m,we identify explicitlythe conditionsunder whichthe mth iterative algorithmconverges to this fixed point.Finally,we show thatwhenthe iterationis unstable(orconverges slowly)it may be stabilized(orits convergencemay be accelerated)by applicationof the Recursive Projection Method.Alternatively,the Newton-KrylovGeneralized Minimal Residual Methodmay be used.In the subsequent article,we will considerthe accuracy and convergenceof the iterative algorithmsfor a broader classof systems – in whichthere need not bean explicitsmall parameter – to whichthe algorithms also apply.

This paper deals with modeling the passivebehavior of skeletal muscle tissue includingcertain microvibrations at the cell level. Our approach combines a continuum mechanics model with large deformation and incompressibility at the macroscale with chains of coupled nonlinear oscillators.The model verifies that an externally appliedvibration at the appropriate frequency is able to synchronize microvibrations in skeletal muscle cells.From the numerical analysis point of view, one faces here a partial differential-algebraic equation (PDAE) that after semi-discretization in space by finite elements possessesan index up to three, depending on certain physicalparameters. In this context, the consequences forthe time integration as well as possible remedies are discussed.

In this paper, we studythe linear Schrödinger equation over the d-dimensional torus,with small values of the perturbing potential.We consider numerical approximations of the associated solutions obtainedby a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable).In this fully discrete setting, we prove that the regularity of the initialdatum is preserved over long times, i.e. times that are exponentially longwith the time discretization parameter. We here refer to Gevrey regularity, and our estimatesturn out to be uniform in the space discretization parameter.This paper extends [G. Dujardin and E. Faou, Numer. Math.97 (2004) 493–535], where a similar result has been obtained inthe semi-discrete situation, i.e. when the mere time variable is discretized and spaceis kept a continuous variable.

Many complex systems occurring in various application share the property that theunderlying Markov process remains in certain regions of the state space for long times,and that transitions between such metastable sets occur only rarely.Often the dynamics within each metastable set is of minor importance, but the transitions between these sets are crucial for the behavior and the understanding of the system.Since simulations of the original process are usually prohibitively expensive, the effective dynamics of the system, i.e. the switching between metastable sets, has to be approximated in a reliable way.This is usually done by computing the dominant eigenvectors and eigenvalues of the transferoperator associated to the Markov process. In many real applications, however, the matrix representing the spatially discretized transfer operator can be extremely large, such that approximating eigenvectors and eigenvaluesis a computationally critical problem.In this article we present a novel method to determine the effective dynamics via the transfer operator without computing its dominant spectral elements. The main idea is that a time series of the process allows to approximate the sampling kernel of the process, which is an integral kernel closely related to the transition function of the transfer operator.Metastability is taken into account by representing the approximative sampling kernel by a linear combination of kernels each of which represents the process on one of the metastable sets.The effect of the approximation error on the dynamics of the system is discussed,and the potential of the new approach is illustrated by numerical examples.

The integration to steady state of many initial value ODEs and PDEs using the forward Euler methodcan alternatively be considered as gradient descent for an associated minimization problem.Greedy algorithms such as steepest descent for determining the step size are asslow to reach steady state as is forward Euler integration with the best uniform step size.But other, much faster methods using bolder step size selection exist.Various alternatives are investigated from both theoretical and practical points of view.The steepest descent method is also known for the regularizing or smoothing effect that thefirst few steps have for certain inverse problems,amounting to a finite time regularization. We further investigate the retention of thisproperty using the faster gradient descent variants in the context of two applications.When the combination of regularization and accuracy demands more than a dozen or so steepestdescent steps, the alternatives offer an advantage, even though (indeed because)the absolute stability limit of forward Euler is carefully yet severely violated.

We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.

This paper deals with topology optimization of domains subject to a pointwise constraint on the gradient of the state. To realize this constraint, a class of penalty functionals is introduced and the expression of the corresponding topological derivative is obtained for the Laplace equation in two space dimensions. An algorithm based on these concepts is proposed. It is illustrated by some numerical applications.

In terms of the normal cone and the coderivative,we provide some necessary and/or sufficient conditions of metric subregularity for(not necessarily closed) convex multifunctions in normed spaces. As applications, we present someerror bound results for (not necessarily lower semicontinuous) convex functions on normedspaces. These results improve and extend some existing error bound results.

A corrected version of [P. Grabowski and F.M. Callier, ESAIM: COCV12 (2006) 169–197], Theorem 4.1, p. 186, and Example, is given.

The paper studies optimal portfolio selection for discrete timemarket models in mean-variance and goal achieving setting. Theoptimal strategies are obtained for models with an observed processthat causes serial correlations of price changes. The optimalstrategies are found to be myopic for the goal-achieving problem andquasi-myopic for the mean variance portfolio.

This paper is devoted to an analysis of vortex-nucleationfor a Ginzburg-Landau functional withdiscontinuous constraint. This functional has been proposedas a model for vortex-pinning, and usuallyaccounts for the energyresulting from the interface of two superconductors. Thecritical applied magnetic field for vortex nucleation is estimated inthe London singular limit,and as a by-product, results concerning vortex-pinning andboundary conditions on the interface are obtained.

We present a phase field approach to wetting problems, related tothe minimization of capillary energy. We discuss in detail boththe Γ-convergence results on which our numerical algorithmare based, and numerical implementation. Two possible choices ofboundary conditions, needed to recover Young's law for the contactangle, are presented. We also consider an extension of theclassical theory of capillarity, in which the introduction of adissipation mechanism can explain and predict the hysteresis ofthe contact angle. We illustrate the performance of the model byreproducing numerically a broad spectrum of experimental results:advancing and receding drops, drops on inclined planes andsuperhydrophobic surfaces.

We derive a posteriori estimates for a discretization in space of the standardCahn–Hilliard equation with a double obstacle free energy.The derived estimates are robust and efficient, and in practice are combinedwith a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compareour method with an existing heuristic spatial mesh adaptation algorithm.

In this paper, we study a postprocessing procedure for improvingaccuracy of the finite volume element approximations of semilinearparabolic problems. The procedure amounts to solve a source problemon a coarser grid and then solve a linear elliptic problem on afiner grid after the time evolution is finished. We derive errorestimates in the L 2 and H 1 norms for the standard finitevolume element scheme and an improved error estimate in the H 1 norm. Numerical results demonstrate the accuracy and efficiency ofthe procedure.

This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdotf(x,u)=0$ on a closed Riemannian manifold M.For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$

We derive a posteriori error estimates for singularlyperturbed reaction–diffusion problems which yield a guaranteedupper bound on the discretization error and are fully and easilycomputable. Moreover, they are also locally efficient and robust inthe sense that they represent local lower bounds for the actualerror, up to a generic constant independent in particular of thereaction coefficient. We present our results in the framework ofthe vertex-centered finite volume method but their nature isgeneral for any conforming method, like the piecewise linear finiteelement one. Our estimates are based on a H(div)-conformingreconstruction of the diffusive flux in the lowest-orderRaviart–Thomas–Nédélec space linked with mesh dual to the originalsimplicial one, previously introduced by the last author in thepure diffusion case. They also rely on elaborated Poincaré,Friedrichs, and trace inequalities-based auxiliary estimatesdesigned to cope optimally with the reaction dominance. In order tobring down the ratio of the estimated and actual overall energyerror as close as possible to the optimal value of one,independently of the size of the reaction coefficient, we finallydevelop the ideas of local minimizations of the estimators by localmodifications of the reconstructed diffusive flux. The numericalexperiments presented confirm the guaranteed upper bound,robustness, and excellent efficiency of the derived estimates.