In this paper, we investigate the coupling between operator splitting techniques and a timeparallelization scheme, the parareal algorithm,as a numericalstrategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts,spatially very localized.In a series of previous studies, the numerical analysis of theoperator splitting as well as the parareal algorithmhas been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterizationof such techniques for these stiff configurations.Therefore,we conduct in this work a precise numerical analysis that considers thecombination of time operator splitting and the parareal algorithmin the context of stiff reaction fronts. The impact of the stiffness featured by these frontson the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution ofsuch problems.We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimatesand examine the performance of such strategiesin the context of academical one-dimensional test casesas well as multi-dimensional configurationssimulated on parallel architecture.

The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions,depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000,Springer (2010)], the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The present paper complements this work. It is shown that one can reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.

We study an atomistic pair potential-energy E (n)(y) that describesthe elastic behavior of two-dimensional crystals with n atoms where $y \in {\mathbb R}^{2\times n}$ characterizes the particle positions. The mainfocus is the asymptotic analysis of the ground state energy as ntends to infinity. We show in a suitable scaling regime where theenergy is essentially quadratic that the energy minimum of E (n)admits an asymptotic expansion involving fractional powers of n:

${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$

The bulk energy density E bulk is given by an explicitexpression involving the interaction potentials. The surface energyE surface can be expressed as a surface integral where theintegrand depends only on the surface normal and the interactionpotentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggestthat the integrand is a continuous, but nowhere differentiable function ofthe surface normal.

Several realistic situations in vehicular traffic that give rise to queues can be modeled through conservation laws with boundary and unilateral constraints on the flux. This paper provides a rigorous analytical framework for these descriptions, comprising stability with respect to the initial data, to the boundary inflow and to the constraint. We present a framework to rigorously state optimal management problems and prove the existence of the corresponding optimal controls. Specific cases are dealt with in detail through ad hoc numerical integrations. These are here obtained implementing the wave front tracking algorithm, which appears to be very precise in computing, for instance, the exit times.

This paper is concerned with the dual formulation of the interface problemconsisting of a linear partial differential equation with variable coefficientsin some bounded Lipschitz domain Ω in $\mathbb{R}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in theunbounded exterior domain Ωc := $\mathbb{R}^n\backslash\bar\Omega$ . The two problems are coupled by transmission andSignorini contact conditions on the interface Γ = ∂Ω. The exterior part of theinterface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequalitywith a linear operator. Then we treat the corresponding numerical scheme and discuss anapproximation of the NtD mapping with an appropriatediscretization of the inverse Poincaré-Steklov operator. In particular, assuming some abstract approximationproperties and a discrete inf-sup condition, we show unique solvability of the discrete scheme andobtain the corresponding a-priori error estimate. Next, we prove that these assumptions aresatisfied with Raviart-Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory.

We prove a posteriori error estimates of optimal order for linearSchrödinger-type equations in the L ∞(L 2)- and theL ∞(H 1)-norm. We discretize only in time by theCrank-Nicolson method. The direct use of the reconstructiontechnique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds thatare of optimal order in the L ∞(L 2)-norm, but ofsuboptimal order in the L ∞(H 1)-norm. The optimality inthe case of L ∞(H 1)-norm is recovered by using anauxiliary initial- and boundary-value problem.

As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.

The Poiseuille flow of a generalized Maxwell fluid is discussed. The velocity field and shear stress corresponding to the flow in an infinite circular cylinder are obtained by means of the Laplace and Hankel transforms. The motion is caused by the infinite cylinder which is under the action of a longitudinal time-dependent shear stress. Both solutions are obtained in the form of infinite series. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases. Finally, the influence of the material and fractional parameters on the fluid motion is brought to light.

An important test of the quality of numerical methods developed to track the interface between two fluids is their ability to reproduce test cases or benchmarks. However, benchmark solutions are scarce and virtually nonexistent for complex geometries. We propose a simple method to generate benchmark solutions in the context of the two-layer flow problem, a classical multiphase flow problem. The solutions are obtained by considering the inverse problem of finding the required channel geometry to obtain a prescribed interface profile. This viewpoint shift transforms the problem from that of having to solve a complex differential equation to the much easier one of finding the roots of a quartic polynomial.

We propose a new primal-dual interior-point algorithm based on a new kernel function for linear optimization problems. New search directions and proximity functions are proposed based on the kernel function. We show that the new algorithm has and iteration bounds for large-update and small-update methods, respectively, which are currently the best known bounds for such methods.

This article considers the linear 1-d Schrödinger equation in (0,π)perturbed by a vanishing viscosity term depending on a small parameterε > 0. We study the boundary controllability properties of thisperturbed equation and the behavior of its boundary controlsvε as ε goes to zero. Itis shown that, for any time T sufficiently large but independent ofε and for each initial datum inH−1(0,π), there exists a uniformly boundedfamily of controls(vε)ε inL2(0, T) acting on the extremityx = π. Any weak limit of this family is a control forthe Schrödinger equation.