On the locking phenomenon for a class of elliptic problems

Summary. In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not. Revised version received October 25, 1993     

Stability and Asymptotic Behaviour for the Numerical Solution of a Reaction--Diffusion Model for a Deterministic Diffusive Epidemic

The aim of this paper is to outline the numerical solution ofa reaction—diffusion system describing the evolution ofan epidemic in an isolated habitat. The model we consider isdescribed by two weakly coupled semi-linear parabolic equationsand we introduce a finite difference scheme for its numericalsolution. We study the behaviour of the exact solution by meansof the numerical scheme. We show the positivity, the decreaseand the decay to extinction of the numerical solution. Finallywe report the results of the numerical tests; in these simulationswe observe that the asymptotic behaviour of the reaction-diffusionsystem is the same as that of the associated ODE system (Kermack—McKendrickmodel).     

Viscous fluid flow at all speeds: analysis and numerical simulation

In [43] a finite volume method for reliable simulations of inviscid fluid flows at high as well as low Mach numbers based on a preconditioning technique proposed by Guillard and Viozat [14] is presented. In this paper we describe an extension of the numerical scheme for computing solutions of the Euler and Navier-Stokes equations. At first we show the high resolution properties, accuracy and robustness of the finite volume scheme in the context of a wide range of complicated transonic and supersonic test cases whereby both inviscid and viscous flow fields are considered. Thereafter, the validity of the method in the low Mach number regime is proven by means of an asymptotic analysis as well as numerical simulations. Whereas in [43] the asymptotic analysis of the scheme is focused on the behaviour of the continuous and discrete pressure distribution for inviscid low speed simulations we prove both the physical sensible discrete pressure field for viscous low Mach number flows and the divergence free condition of the discrete velocity field in the limit of a vanishing Mach number with respect to the simulation of inviscid fluid flow.     

Viscous fluid flow at all speeds: analysis and numerical simulation

In [43] a finite volume method for reliable simulations of inviscid fluid flows at high as well as low Mach numbers based on a preconditioning technique proposed by Guillard and Viozat [14] is presented. In this paper we describe an extension of the numerical scheme for computing solutions of the Euler and Navier-Stokes equations. At first we show the high resolution properties, accuracy and robustness of the finite volume scheme in the context of a wide range of complicated transonic and supersonic test cases whereby both inviscid and viscous flow fields are considered. Thereafter, the validity of the method in the low Mach number regime is proven by means of an asymptotic analysis as well as numerical simulations. Whereas in [43] the asymptotic analysis of the scheme is focused on the behaviour of the continuous and discrete pressure distribution for inviscid low speed simulations we prove both the physical sensible discrete pressure field for viscous low Mach number flows and the divergence free condition of the discrete velocity field in the limit of a vanishing Mach number with respect to the simulation of inviscid fluid flow.     

Adaptive timestep control for the contact-stabilized Newmark method

The aim of this paper is to devise an adaptive timestep control in the contact-stabilized Newmark method (ContacX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini’s condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact-stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.     

A parameter robust numerical method for a two dimensional reaction-diffusion problem

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.

     

A polyhedral study on 0–1 knapsack problems with disjoint cardinality constraints: Strong valid inequalities by sequence-independent lifting

We study the set of 0–1 integer solutions to a single knapsack constraint and a set of non-overlapping cardinality constraints (MCKP), which generalizes the classical 0–1 knapsack polytope and the 0–1 knapsack polytope with generalized upper bounds. We derive strong valid inequalities for the convex hull of its feasible solutions using sequence-independent lifting. For problems with a single cardinality constraint, we derive two-dimensional superadditive lifting functions and prove that they are maximal and non-dominated under some mild conditions. We then show that these functions can be used to build strong valid inequalities for problems with multiple disjoint cardinality constraints. Finally, we present preliminary computational results aimed at evaluating the strength of the cuts obtained from sequence-independent lifting with respect to those obtained from sequential lifting.     

Stability of the numerical solution of the periodic reaction- diffusion systems

In this paper we study the behavior of the numerical solution of nonlinear reaction-diffusion systems, with periodic in time nonlinear term, obtained via the known θ-method. In particular we are interested to the existence and asymptotic stability of the numerical periodic solutions in order to simulate the behaviour of the theoretical solution. To this end, by imposing the positivity of the numerical scheme, we can use some results about M-matrices.So, by means of particular over and upper solutions, we study some conditions for the stability and instability of the trivial solution.Finally we show when a positive numerical periodic solution exists and when it is unique and asymptotically stable.     

Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints

We study perturbations of a stochastic program with a probabilistic constraint and r-concave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distance of probability distributions. Secondly, we derive an upper Lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures. The latter analysis applies to linear-quadratic models and is based on work by Bonnans and Shapiro. The stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint.Mathematics Subject Classification (2000): 90C15, 90C31     

Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.     

Dissipativity of the linearly implicit Euler scheme for Navier‐Stokes equations with delay

In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier‐Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L² dissipative for any time step size and H¹ dissipative under a time‐step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by‐product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit‐explicit scheme for the celebrated Navier‐Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017     

Interface cracks in anisotropic composites

The linear model equations of elasticity often give rise to oscillatory solutions in some vicinity of interface crack fronts. In this paper we apply the Wiener–Hopf method which yields the asymptotic behaviour of the elastic fields and, in addition, criteria to prevent oscillatory solutions. The exponents of the asymptotic expansions are found as eigenvalues of the symbol of corresponding boundary pseudodifferential equations. The method works for three‐dimensional anisotropic bodies and we demonstrate it for the example of two anisotropic bodies, one of which is bounded and the other one is its exterior complement. The common boundary is a smooth surface. On one part of this surface, called the interface, the bodies are bonded, while on the complementary part there is a crack. By applying the potential method, the problem is reduced to an equivalent system of Boundary Pseudodifferential Equations (BPE) on the interface with the stress vector as the unknown. The BPEs are defined via Poincaré–Steklov operators. We prove the unique solvability of these BPEs and obtain the full asymptotic expansion of the solution near the crack front. As a special case we consider the interface crack between two different isotropic materials and derive an explicit criterion which prevents oscillatory solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Efficient integration of flexible multibody dynamics

The modelling of flexible multibody dynamics as finite dimensional Hamiltonian system subject to holonomic constraints constitutes a general framework for a unified treatment of rigid and elastic components. Internal constraints, which are associated with the kinematic assumptions of the underlying continuous theory, as well as external constraints, representing the interconnection of different bodies by joints, can be accounted for in a likewise systematic way. The discrete null space method developed in [0] provides an energy-momentum conserving integration scheme for the DAEs of motion of constrained mechanical systems. It relies on the elimination of the constraint forces from the discrete system along with a reparametrisation of the nodal unknowns. The resulting reduced scheme performs advantageously concerning different aspects: the constraints are fulfilled exactly, the condition number of the iteration matrix is independent of the time step and the dimension of the system is reduced to the minimal possible number saving computational costs. A six-body-linkage possessing a single degree of freedom is analysed as an example of a closed loop structure. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A uniformly convergent scheme for a system of reaction–diffusion equations

In this work a system of two parabolic singularly perturbed equations of reaction–diffusion type is considered. The asymptotic behaviour of the solution and its partial derivatives is given. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution we consider the implicit Euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform Shishkin mesh. We prove that this scheme is uniformly convergent, with respect to the diffusion parameters, having first-order convergence in time and almost second-order convergence in space, in the discrete maximum norm. Numerical experiments illustrate the order of convergence proved theoretically.     

椭圆-抛物偏微分方程奇异摄动问题的差分解法

本文讨论了含有小参数在高阶导数项的椭圆型方程奇异摄动问题的差分解法.当ε=0时椭圆型方程退化为抛物型方程.作者根据此问题解的边界层性质,构造了特殊的差分格式:研究了它的收敛性和解的渐近性态.最后给出一个数值例题.     

Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.     

Incompressible non‐Newtonian fluids: time asymptotic behaviour of weak solutions

We study the asymptotic behaviour in time of incompressible non‐Newtonian fluids in the whole space assuming that initial data also belong to L¹. Firstly, we consider the weak solution to the power‐law model with non‐zero external forces and we find the asymptotic behaviour in time of this solution in the same class of existence and uniqueness with p?. Secondly, we are interested in the asymptotic behaviour of weak solutions to the second grade model, and finally, we deal with the asymptotic behaviour in time of weak solutions to a simplified model of viscoelastic fluids of the Oldroyd type. Copyright © 2006 John Wiley & Sons, Ltd.     

Asymptotic analysis of a thin‐layer device with Tresca's contact law in elasticity

In this paper, we consider a thin elastic layer between a rigid body and an elastic one. A Tresca law is assumed between the two elastic bodies. The Lamé coefficients of the thin layer are assumed to vary with respect to its height ϵ. This dependence is shown to be of primary importance in the asymptotic behaviour of the device, a critical case leading to a non‐classical contact law when deleting the bond. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical resolution of a mono-disperse model of bubble growth in magmas

Growth of gas bubbles in magmas may be modeled by a system of differential equations that account for the evolution of bubble radius and internal pressure and that are coupled with an advection–diffusion equation defining the gas flux going from magma to bubble. This system of equations is characterized by two relaxation parameters linked to the viscosity of the magma and to the diffusivity of the dissolved gas, respectively. Here, we propose a numerical scheme preserving, by construction, the total mass of water of the system. We also study the asymptotic behavior of the system of equations by letting the relaxation parameters vary from 0 to ∞, and show the numerical convergence of the solutions obtained by means of the general numerical scheme to the simplified asymptotic limits. Finally, we validate and compare our numerical results with those obtained in experiments.     

High‐order asymptotic‐preserving schemes for linear systems: Application to the Goldstein–Taylor equations

In this work, we address the numerical approximation of linear systems with possibly stiff source terms which induce an asymptotic diffusion limit. More precisely, we are interested in the design of high‐order asymptotic‐preserving schemes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme. This alteration can be understood as a numerical diffusion reduction technique and allows to capture the correct asymptotic behavior in the diffusion limit and to consider uniformly high‐order extensions. We more specifically consider the case of the Goldstein–Taylor model but the overall approach is shown to be easily adapted to more general systems.