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Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods., Comment: 25 pages, 2 figures, 2 tables

In this paper we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some numerical tests are reported, in order to assess the theoretical findings., 30 pages, 5 figures, 3 tables

In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than that using the expression of the polynomials w.r.t. the canonical basis. This algorithm is aimed at solving corresponding fractional differential equations. A few numerical examples are reported., Comment: 8 pages, 3 figures

Recently, the numerical solution of stiffly/highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. While a theoretical analysis of this spectral approach has been only partially addressed, there is enough numerical evidence that it turns out to be very effective even when applied to a wider range of problems. Here we fill this gap by providing a thorough convergence analysis of the methods and confirm the theoretical results with the aid of a few numerical tests., 18 pages, 1 figure

We implement the PaperRank and AuthorRank indices introduced in [Amodio & Brugnano, 2014] in the Scopus database, in order to highlight quantitative and qualitative information that the bare number of citations and/or the h-index of an author are unable to provide. In addition to this, the new indices can be cheaply updated in Scopus, since this has a cost comparable to that of updating the number of citations. Some examples are reported to provide insight in their potentialities, as well as possible extensions., 20 pages, 3 figures, 4 tables. Accepted version on Journal of Infometrics

We describe the analytical solution of the eigenvalue problem introduced in our article mentioned in the title and relative to a punctiform electric charge confined in an one-dimensional box in the presence of an electric field. We also derive and discuss the analytical expressions of the external forces acting on the punctiform charge and associated with the boundaries of the one-dimensional box in the presence of the electric field.

We introduce a dynamic precision floating-point arithmetic based on the Infinity Computer. This latter is a computational platform which can handle both infinite and infinitesimal quantities epitomized by the positive and negative finite powers of the symbol Open image in new window, which acts as a radix in a corresponding positional numeral system. The idea is to use the positional numeral system from the Infinity Computer to devise a variable precision representation of finite floating-point numbers and to execute arithmetical operations between them using the Infinity Computer Arithmetics. Here, numerals with negative finite powers of Open image in new window will act as infinitesimal-like quantities whose aim is to dynamically improve the accuracy of representation only when needed during the execution of a computation. An application to the iterative refinement technique to solve linear systems is also presented.

We focus on the solution of multiparameter spectral problems, and in particular on some strategies to compute coarse approximations of selected eigenparameters depending on the number of oscillations of the associated eigenfunctions. Since the computation of the eigenparameters is crucial in codes for multiparameter problems based on finite differences, we herein present two strategies. The first one is an iterative algorithm computing solutions as limit of a set of decoupled problems (much easier to solve). The second one solves problems depending on a parameter \(\sigma \in [0,1]\), that give back the original problem only when \(\sigma =1\). We compare the strategies by using well known test problems with two and three parameters.

We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities symbolized by the positive and negative finite powers of the radix grossone. The computational features offered by the Infinity Computer allows us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented., Comment: 11 pages, 2 figures, 6 tables

Recently, Hamiltonian Boundary Value Methods (HBVMs), have been used as spectral methods in time for effectively solving multi-frequency, highly-oscillatory and/or stiffly-oscillatory problems. A complete analysis of their use in such a fashion has been also carried out, providing a theoretical framework explaining their effectiveness. We report here a few numerical examples showing their potentialities to provide a fully accurate solver for general ODE problems.

In this paper, we consider a regular grid with equal spatial spacings and construct a new finite difference approximation (difference scheme) for the system of two-dimensional Navier–Stokes equations describing the unsteady motion of an incompressible viscous liquid of constant viscosity. In so doing, we use earlier constructed discretization of the system of three equations: the continuity equation and the proper Navier–Stokes equations. Then, we compute the canonical Grobner basis form for the obtained discrete system. It gives one more difference equation which is equivalent to the pressure Poisson equation modulo difference ideal generated by the Navier–Stokes equations, and thereby comprises a new finite difference approximation (scheme). We show that the new scheme is strongly consistent. Besides, our computational experiments demonstrate much better numerical behavior of the new scheme in comparison with the other strongly consistent schemes we constructed earlier and with the scheme which is not strongly consistent.

In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is the Newton–Raphson technique combined with the quasi-time marching. In the first step, a solution of a simpler problem, as an initial guess for the Newton–Raphson iterations, is provided. Then, step-by-step, this simpler problem is converted into the original problem, while the quasi-time parameter τ runs from τ = 0 to τ = 1 . While following the involved imaginary path two numerical approaches are realized, the first is based on the Prufer angle technique, the second on high order finite difference schemes.

We describe a study motivated by our interest to examine the incompleteness of the Ehrenfest's theorem in quantum mechanics and to resolve a doubt regarding whether or not the hermiticity of the hamiltonian operator is sufficient to justify a simplification of the expression of the macroscopic-observable time derivative that promotes the one usually found in quantum-mechanics textbooks. The study develops by considering the simple quantum system "particle in one-dimensional box". We propose theoretical arguments to support the incompleteness of the Ehrenfest's theorem in the formulation he gave, in agreement with similar findings already published by a few authors, and corroborate them with the numerical example of an electric charge in an electrostatic field. The contents of this study should be useful to Bachelor and Master students; the style of the discussions is tailored to stimulate, we hope, the student's ability for independent thinking., 35 pages, 5 figures, 1 table. Fixed typo in Ref. [15]. Published in the European Journal of Physics (https://iopscience.iop.org/journal/0143-0807) on 16 September 2021 and updated on 14 October 2021

A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer—a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of y ( t ) (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four.

In recent years, the class of energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) has been devised for numerically solving Hamiltonian problems. In this short note, we study their natural formulation as continuous-stage Runge-Kutta(-Nystr\"om) methods, which allows a deeper insight in the methods., Comment: 12 pages, accepted manuscript

In the present paper we analyze and discuss some mathematical aspects of the fluid-static configurations of a self-gravitating perfect gas enclosed in a spherical solid shell. The mathematical model we consider is based on the well-known Lane-Emden equation, albeit under boundary conditions that differ from those usually assumed in the astrophysical literature. The existence of multiple solutions requires particular attention in devising appropriate numerical schemes apt to deal with and catch the solution multiplicity as efficiently and accurately as possible. In sequence, we describe some analytical properties of the model, the two algorithms used to obtain numerical solutions, and the numerical results for two selected cases., Comment: Revision accepted for publication in "Communications in Computational Physics"; 23 pages, 9 figures

In this note, we extend a technique recently used to devise a novel class of geometric integrators named Hamil-tonian Boundary Value Methods, to cope with nonlinear fractional differential equations. The approach relies on a truncated Fourier expansion of the vector field which yields a modified problem that can be suitably handled on a computer. An example showing the convergence properties of the resulting spectral approximation method is also presented.

We open the paper with introductory considerations describing the motivations of our long-term research plan targeting gravitomagnetism, illustrating the fluid-dynamics numerical test case selected for that purpose, that is, a perfect-gas sphere contained in a solid shell located in empty space sufficiently away from other masses, and defining the main objective of this study: the determination of the gravitofluid-static field required as initial field ($t=0$) in forthcoming fluid-dynamics calculations. The determination of the gravitofluid-static field requires the solution of the isothermal-sphere Lane-Emden equation. We do not follow the habitual approach of the literature based on the prescription of the central density as boundary condition; we impose the gravitational field at the solid-shell internal wall. As the discourse develops, we point out differences and similarities between the literature's and our approach. We show that the nondimensional formulation of the problem hinges on a unique physical characteristic number that we call gravitational number because it gauges the self-gravity effects on the gas' fluid statics. We illustrate and discuss numerical results; some peculiarities, such as gravitational-number upper bound and multiple solutions, lead us to investigate the thermodynamics of the physical system, particularly entropy and energy, and preliminarily explore whether or not thermodynamic-stability reasons could provide justification for either selection or exclusion of multiple solutions. We close the paper with a summary of the present study in which we draw conclusions and describe future work., Comment: 32 pages, 26 figures

In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Korteweg–de Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value p = 5 , solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in Klein and Peter (2015) indicate that if p > 5 the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at p = 5 . Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as p → 5 + . By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods.

We discuss the solution of regular and singular Sturm–Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm–Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.

In this paper, we discuss the progress in the numerical simulation of the so-called ‘whispering gallery’ modes (WGMs) occurring inside a prolate spheroidal cavity. These modes are mainly concentrated in a narrow domain along the equatorial line of a spheroid and they are famous because of their extremely high quality factor. The scalar Helmholtz equation provides a sufficient accuracy for WGM simulation and (in a contrary to its vector version) is separable in spheroidal coordinates. However, the numerical simulation of ‘whispering gallery’ phenomena is not straightforward. The separation of variables yields two spheroidal wave ordinary differential equations (ODEs), first only depending on the angular, second on the radial coordinate. Though separated, these equations remain coupled through the separation constant and the eigenfrequency, so that together with the boundary conditions they form a singular self-adjoint two-parameter Sturm–Liouville problem. We discuss an efficient and reliable technique for the numerical solution of this problem which enables calculation of highly localized WGMs inside a spheroid. The presented approach is also applicable to other separable geometries. We illustrate the performance of the method by means of numerical experiments.

In this work, we discuss the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. Due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we present several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.

New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial conditions are proposed. They are designed for working on a new kind of a supercomputer – the Infinity Computer – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods are able to work with the exact values of the derivatives, instead of their approximations. Within this context, variants of one-step multi-point methods closely related to the classical Taylor formulae and to the Obrechkoff methods are considered. To get numerical evidence of the theoretical results, test problems are solved by means of the new methods and the results compared with the performance of classical methods.

We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spectral problems for ordinary differential equations. We describe how to obtain a discrete problem by means of High Order Finite Difference Schemes and discuss its numerical solution. Based on this approach, we also define a recursive algorithm to compute approximations of the parameters by means of the solution of a set of problems converging to the original one.

We show the main features of the MATLAB code HOFiD_UP for solving second order singular perturbation problems. The code is based on high order finite differences, in particular on the generalized upwind method. Within its simplicity, it uses order variation and continuation for solving any difficult nonlinear scalar problem. Several numerical tests on linear and nonlinear problems are considered. The best performances are reported on problems with perturbation parameters near the machine precision, where most of the codes for two-point BVPs fail.

The parallel solution of initial value problems for ordinary differential equations (ODE-IVPs) has received much interest from many researchers in the past years. In general, the possibility of using parallel computing in this setting concerns different aspects of the numerical solution of ODEs, depending on the parallel platform to be used and/or the complexity of the problem to be solved. In particular, in this paper we examine possible extensions of a parallel method previously proposed in the mid-nineties [P. Amodio, L. Brugnano, Parallel implementation of block boundary value methods for ODEs, J. Comput. Appl. Math. 78 (1997) 197-211; P. Amodio, L. Brugnano, Parallel ODE solvers based on block BVMs, Adv. Comput. Math. 7 (1997) 5-26], and analyze its connections with subsequent approaches to the parallel solution of ODE-IVPs, in particular the ''Parareal'' algorithm proposed in [J.L. Lions, Y. Maday, G. Turinici, Resolution d'EDP par un schema en temps ''parareel'', C. R. Acad. Sci. Paris, Ser. I 332 (2001) 661-668; Y. Maday, G. Turinici, A parareal in time procedure for the control of partial differential equations, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 387-392].

High even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation problems are taken into account to emphasize the main features of the proposed methods.

We propose an efficient and reliable technique to calculate highly localized Whispering Gallery Modes (WGMs) inside an oblate spheroidal cavity. The idea is to first separate variables in spheroidal coordinates and then to deal with two ODEs, related to the angular and radial coordinates solved using high order finite difference schemes. It turns out that, due to solution structure, the efficiency of the calculation is greatly enhanced by using variable stepsizes to better reflect the behaviour of the evaluated functions. We illustrate the approach by numerical experiments.

BABDCR is a package of Fortran 90 subroutines for the solution of linear systems with bordered almost block diagonal coefficient matrices. It is designed to handle matrices with blocks of the same size, that is, having a block upper bidiagonal structure with an additional block in the right upper corner. The algorithm implemented in the package performs cyclic reduction of the coefficient matrix in order to reduce the fill-in due to the corner block.

We introduce symmetric Boundary Value Methods for the solution of second order initial and boundary value problems (in particular Hamiltonian problems). We study the conditioning of the methods and link it to the boundary loci of the roots of the associated characteristic polynomial. Some numerical tests are provided to assess their reliability.

Given a Hamiltonian matrix H = JS with S symmetric and positive definite, we analyze a symplectic Lanczos algorithm to transform - H2 in a symmetric and positive definite tridiagonal matrix of half size. By means of two effective restarted procedures, this algorithm is then used to compute few extreme eigenvalues of H. Numerical examples are also reported to compare the presented techniques.

We analyse the behaviour of symmetric boundary value methods applied to Hamiltonian evolutionary problems. We derive a forward–backward error analysis which allows us to state that the obtained numerical solution may be essentially regarded as the exact solution of a perturbed Hamiltonian problem of the same type as the original one and vice versa. A numerical test emphasizes the theoretical results. Copyright © 2004 John Wiley & Sons, Ltd.

We consider Hamiltonian matrices obtained by means of symmetric and positive definite matrices and analyse some perturbations that maintain the eigenvalues on the imaginary axis of the complex plane. To obtain this result we prove for such matrices the existence of a diagonal form or, alternatively by means of symplectic transformations, the existence of the simplest canonical form. Applications related to a pair of problems in the context of linear algebra and differential equations are also reported. Copyright © 2004 John Wiley & Sons, Ltd.

In this work, two different extraction procedures for the analysis of different polycyclic aromatic hydrocarbons (PAHs) in water by microextraction by packed sorbent (MEPS) have been compared in terms of sensitivity, reliability and time of analysis. The first method, called "draw-eject", consists of a sequence of cycles of aspirations and injections in the same vial; the second one, called "extract-discard", consists of a similar cycle sequence, but the aspired sample in this case is discarded into waste. The relevant partition equilibriums and extraction rates have been calculated by multivariate regression from the data obtained after MEPS gas chromatography-mass spectrometry (MEPS-GC-MS) analysis of 16 PAHs from water samples. Partitioning parameters for a priori prediction of solute sorption equilibrium, recoveries and preconcentration effects in aqueous and solvent systems have been calculated and compared for the two extraction procedures. Finally, real samples from sea, agricultural irrigation wells, streams and tap water were analyzed. Detection (S/N≥3) and quantification (S/N≥10) limits were calculated for the extraction processes. Under the experimental conditions used for the "draw-eject" procedure, these values were in the range 0.5-2 ng L(-1) and 1.6-6.2 ng L(-1), while for the "extract-discard" procedure they ranged from 0.2 to 0.8 ng L(-1) and from 0.8 to 2.0 ng L(-1), respectively.

We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in saturated porous media such as concrete. Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy.

We discuss the solution of regular and singular Sturm–Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm–Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.

A new backward error analysis of LU factorization is presented. It allows to obtain a sharper upper bound for the forward error and a new definition of the growth factor that we compare with the well known Wilkinson growth factor for some classes of matrices. Numerical experiments show that the new growth factor is often of order approximately log2n whereas Wilkinson's growth factor is of order n or \(\sqrt n\).