1,721,186 research outputs found
Efficient computation of the sinc matrix function for the integration of second-order differential equations
No full textThis work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit (pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on sinc matrix functions. The novelty contained here is that of using a suitable rational approximation formula for the sinc matrix function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we discuss the application of the whole strategy to a finite element discretization of the wave equation
Theoretical error estimates for computing the matrix logarithm by Padé-type approximants
No full textIn this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss–Legendre quadrature rules. These formulas can be interpreted as Padé approximants of a suitable Gauss hypergeometric function. Empirical observation tells us that the convergence of these quadratures becomes slow when the matrix is not close to the identity matrix, thus suggesting the usage of an inverse scaling and squaring approach for obtaining a matrix with this property. The novelty of this work is the introduction of error estimates that can be used to select a priori both the number of Legendre points needed to obtain a given accuracy and the number of inverse scaling and squaring to be performed. We include some numerical experiments to show the reliability of the estimates introduced
Il problema della stabilità per metodi numerici per ODEs
Full text linkIn this paper we introduce and analyze some relations between the Pascal matrix and a new class of numerical methods for differential equations obtained generalizing the Adams methods. In particular, we shall prove that these methods are suitable for solving stiff problems since their absolute stability regions contain the negative half complex plane
An algebraic procedure for the spectral corrections using the miss-distance functions in regular and singular Sturm-Liouville problems
No full textA general method based on the evaluation of the zeros of a suitable polynomial is suggested in order to have an estimation of the spectral error in the numerical treatment of Sturm-Liouville problems. The method is strictly concerned with the miss-distance function arising in the shooting algorithm for eigenvalues. The error correcting procedure derived from the method is particularly helpful when difficulties arise in the numerical integration. Two kinds of Sturm-Liouville problems are considered: the standard regular problems on a closed interval and the problems where an eigenvalue is nonlinearly involved and embedded in an essential spectrum giving origin to an inner singularity. Numerical experiments clearly highlight the efficaciousness of the proposed method both in the regular and singular case
BVMs for Sturm-Liouville eigenvalue estimates with general boundary conditions
No full textRecently, a class of Boundary Value Methods (BVMs) has been introduced for the estimation of the eigenvalues of Sturm-Liouville problems with Dirichlet boundary conditions. The aim of this paper is to extend the application of such BVMs to problems with boundary conditions of general form and to compare the approximations obtained with those given by the corrected Numerov method
A Generalization of Numerov's Method Using the BVM Approach for Sturm-Liouville Eigenvalue estimates
No full textBoundary Value Methods generalizing the Numerov's method are here proposed for the numerical approximation of the eigenvalues of regular Sturm-Liouville problems subject to Dirichlet boundary conditions. Moreover, an analysis of the error in the approximation of the k-th eigenvalue provided by such methods is reported. Some numerical results showing the possible advantages that may arise from the use of the new schemes are also presented
On a generalization of time-accurate and highly-stable explicit operators for stiff problems
No full textIn this work we propose a generalization of the family of Time-Accurate and highly-Stable Explicit (TASE) operators recently introduced by Calvo, Montijano and Rández (2021). In this family the TASE operator of order p depends on p free real parameters. Here we consider operators that can also be defined by complex parameters occurring in conjugate pairs. Despite this choice the calculations continue to be done only in real arithmetic, thus not burdening the computational cost of the previous version of the family. Conversely, this generalization leads to improve both the accuracy and stability properties of explicit Runge Kutta schemes supplemented with TASE operators. Numerical experiments showing the competitiveness of the methods proposed in this paper with respect to classical integrators for stiff problems are also presented
Matrix methods for radial Schroedinger eigenproblems defined on a semi-infinite domain.
Full text linkIn this paper, we discuss numerical approximation of the eigenvalues of
the one-dimensional radial Schr ̈odinger equation posed on a semi-infinite
interval. The original problem is first transformed to one defined on a
finite domain by applying suitable change of the independent variable.
The eigenvalue problem for the resulting differential operator is then ap-
proximated by a generalized algebraic eigenvalue problem arising after
discretization of the analytical problem by the matrix method based on
high order finite difference schemes. Numerical experiments illustrate the
performance of the approach
Boundary Value Methods for the Reconstruction of Sturm-Liouville potentials
No full textThe paper deals with the numerical solution of the two-spectra and the half inverse Sturm-Liouville problems. The numerical procedure proposed provides a continuous approximation of the unknown potential and uses an approach similar to the one studied in [6] for solving the symmetric inverse problem. The results of numerical experiments confirm the effectiveness of the considered methods
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