Remarks on the numerical range with respect to a family of projections

In this note we report on the recently introduced concept of the numerical range of a bounded linear operator on a Hilbert space with respect to a family of projections. The importance of this new concept lies in the fact that it unifies and generalizes well-established versions of the numerical range such as the classical numerical range introduced by Toeplitz and Hausdorff, the quadratic numerical range as well as the block numerical range. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coefficient rings of numerical semigroup algebras

Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show that arithmetic and set-theoretic properties of a numerical semigroup ring are properties of an equi-gcd numerical semigroup algebra.

     

An alternative to the Bathe algorithm

This paper presents a new composite sub-steps algorithm for solving reliable numerical responses in structural dynamics. The newly developed algorithm is a two sub-steps, second-order accurate and unconditionally stable implicit algorithm with the same numerical properties as the Bathe algorithm. The detailed analysis of the stability and numerical accuracy is presented for the new algorithm, which shows that its numerical characteristics are identical to those of the Bathe algorithm. Hence, the new sub-steps scheme could be considered as an alternative to the Bathe algorithm. Meanwhile, the new algorithm possesses the following properties: (a) it produces the same accurate solutions as the Bathe algorithm for solving linear and nonlinear problems; (b) it does not involve any artificial parameters and additional variables, such as the Lagrange multipliers; (c) The identical effective stiffness matrices can be obtained inside two sub-steps; (d) it is a self-starting algorithm. Some numerical experiments are given to show the superiority of the new algorithm and the Bathe algorithm over the dissipative CH-α algorithm and the non-dissipative trapezoidal rule.     

Nonconforming finite element methods without numerical locking

We analyze the numerical approximation of a class of elliptic problems which depend on a small parameter . We give a generalization to the nonconforming case of a recent result established by Chenais and Paumier for a conforming discretization. For both the situations where numerical integration is used or not, a uniform convergence in and h is proved, numerical locking being thus avoided. Important tools in the proof of such a result are compactness properties for nonconforming spaces as well as the passage to the limit problem. Received October 7, 1997     

Mean-square stability analysis of numerical schemes for stochastic differential systems

For ordinary differential systems, the study of A-stability for a numerical method reduces to the scalar case by means of a transformation that uncouples the linear test system as well as the difference system provided by the method. For stochastic differential equations (SDEs), mean-square stability (MS-stability) has been successfully proposed as the generalization of A-stability, and numerical MS-stability has been analyzed for one-dimensional equations. However, unlike the deterministic case, the extension of this analysis to multi-dimensional systems is not straightforward. In this paper we give necessary and sufficient conditions for the MS-stability of multi-dimensional systems with one Wiener noise. The criterion presented does not depend on any norm. Based on the Routh–Hurwitz theorem, we offer a particular criterion of MS-stability for two-dimensional systems in terms of their coefficients. In addition, a counterpart criterion of MS-stability is given for numerical schemes applied to multi-dimensional systems. The MS-stability behavior of a stochastic numerical method is determined by the comparison of its stability region with the stability region of the system. As an application, the numerical MS-stability of θ$\theta$-methods applied to bi-dimensional systems is investigated.     

A new hypothesis concerning children’s fractional knowledge

The basic hypothesis of the teaching experiment, The Child’s Construction of the Rational Numbers of Arithmetic (Steffe & Olive, 1990) was that children’s fractional schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because when a new scheme is established by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. In that case where children’s fractional schemes do emerge as accommodations in their numerical counting schemes, I regard the fractional schemes as superseding their earlier numerical counting schemes. If one scheme supersedes another, that does not mean the earlier scheme is replaced by the superseding scheme. Rather, it means that the superseding scheme solves the problems the earlier scheme solved but solves them better, and it solves new problems the earlier scheme didn’t solve. It is in this sense that we hypothesized children’s fractional schemes can supersede their numerical counting schemes and it is the sense in which we regarded numerical schemes as constructive mechanisms in the production of fractional schemes (Kieren, 1980).     

Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels

We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suffer from a “loss of derivatives”, and that the subsequent linear rate of convergence can be improved to be superlinear using an adaptation of Nash–Moser iteration for numerical analysis purposes; the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also in the inversion step. This revised version was published online in June 2006 with corrections to the Cover Date.     

Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Full discretization of some reaction diffusion equation with blow up

This paper aims at the development of numerical schemes for nonlinear reaction diffusion problems with a convection that blows up in a finite time. A full discretization of this problem that preserves the blow — up property is presented as well as a numerical simulation. Efficiency of the method is derived via a numerical comparison with a classical scheme based on the Runge Kutta scheme.     

On the numerical approximations of stiff convection–diffusion equations in a circle

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a $P_1$ classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.     

Irreducible numerical semigroups with multiplicity three and four

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in Blanco and Puerto (SIAM J. Discrete Math., 26(3):1210–1237, 2012). With this tool we also completely describe the whole family of minimal decompositions into irreducible numerical semigroups with the same multiplicity for this set of numerical semigroups. We give detailed examples to show the applicability of the methodology and conditions for the irreducibility of well-known families of numerical semigroups such as those that are generated by a generalized arithmetic progression.     

Non-isothermal,compressible gas flow for the simulation of an enhanced gas recovery application

In this work, we present a framework for numerical modeling of CO₂ injection into porous media for enhanced gas recovery (EGR) from depleted reservoirs. Physically, we have to deal with non-isothermal, compressible gas flows resulting in a system of coupled non-linear PDEs. We describe the mathematical framework for the underlying balance equations as well as the equations of state for mixing gases. We use an object-oriented finite element method implemented in C++. The numerical model has been tested against an analytical solution for a simplified problem and then applied to CO₂ injection into a real reservoir. Numerical modeling allows to investigate physical phenomena and to predict reservoir pressures as well as temperatures depending on injection scenarios and is therefore a useful tool for applied numerical analysis.     

On the choice of signs for householder's matrices

The numerical construction of Householder's matrices of the form I-2ww^H is known to be a problem with two distinct solutions; more precisely, the actual construction of such a matrix in a given context involves a choice of sign, and it is widely believed that only one alternative is correct, the other one leading to possible numerical unstabilities. This paper shows that the numerical stability of the process depends not on the chosen sign itself but only on the implementation of the actual computations; as well-conditioned approach for the non-classical case is presented and illustrated by a numerical example. Both signs are thus equally correct and there seems to be no reason at all why a specific sign should be prefered to the other.     

High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations

An approach for the creation of high-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations is proposed. New versions of up to the eighth order of accuracy inclusive are implemented. For smooth solutions, numerical experiments on a sequence of grids show that the approximate solutions produced by these versions converge to the exact one with a high order of accuracy as h → 0, where h is the maximal linear cell size of a grid. The numerical results obtained for the benchmark problem of the lid-driven cavity flow suggest that the collocations and least squares method is well suited for the numerical simulation of viscous flows.     

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).     

Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations

In this paper the numerical approximations of the Ginzburg- Landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal Voronoi tessellations. A reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit. We present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations. Properties of the spherical centroidal Voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field.

     

Deficient discrete cubic spline solution for a system of second order boundary value problems

In this paper we use deficient discrete cubic spline to obtain approximate solution of a system of second order boundary value problems. It is shown that the method is of order 2 when a parameter takes a specific value. A well known numerical example is presented to illustrate our method as well as to compare the performance with other numerical methods proposed in the literature.     

Conservative numerical methods for the Full von Kármán plate equations

This article is concerned with the numerical solution of the full dynamical von Kármán plate equations for geometrically nonlinear (large‐amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von Kármán system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semidiscrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretizations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1948–1970, 2015     

A novel numerical approach to simulating nonlinear Schrödinger equations with varying coefficients

We describe a novel numerical approach to simulations of nonlinear Schrödinger equations with varying coefficients, based on the discovery of a new and intrinsic conservation law for varying coefficient nonlinear Schrödinger equations. The approach is shown to preserve some crucial classical conservations, such as the spatial ergodicity, and utilized in numerical simulations of periodically and quasi-periodically solitary waves for nonlinear Schrödinger equations with periodic or quasi-periodic coefficients. Some numerical experiments are presented to illustrate the conservative property.