Cecilia Magherini

Blended implementation of block implicit methods for ODEs

In this paper we further develop a new approach for naturally defining the nonlinear splittings needed for the implementation of block implicit methods for ODEs, which has been considered by Brugnano [J. Comput. Appl. Math. 116 (2000) 41] and by Brugnano and Trigiante [in: Recent Trends in Numerical Analysis, Nova Science, New York, 2000, pp. 81-105]. The basic idea is that of defining the numerical method as the combination (blending) of two suitable component methods. By carefully choosing such methods, it is shown that very efficient implementations can be obtained. Moreover, some of them, characterized by a diagonal splitting, are well suited for parallel computers. Some numerical tests comparing the performances of the proposed implementation with other existing ones are also presented, in order to make evident the potential of the approach.

Boundary Value Methods for the reconstruction of Sturm-Liouville potentials

Abstract In this paper we present numerical procedures for solving the two inverse Sturm–Liouville problems known in the literature as the two-spectra and the half inverse problems. The method proposed looks for a continuous approximation of the unknown potential belonging to a suitable function space of finite dimension. In order to compute such an approximation a sequence of direct problems has to be solved. This is done by applying one of the Boundary Value Methods, generalizing the classical Numerov scheme, recently introduced by the authors. Numerical results confirming the effectiveness of the approach proposed are also reported.

Blended General Linear Methods based on Generalized BDF

General Linear Methods were introduced in order to encompass a large family of numerical methods for the solution of ODE‐IVPs, ranging from LMF to RK formulae. In so doing, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, stability limitations of LMF and order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been done in order to derive GLMs with particular features, to be exploited for their efficient implementation.In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has been proposed for the numerical solution of stiff ODE‐IVPs. Here, this approach is further developed in order to derive GLMs combining good stability and accuracy properties with the possibility of efficiently solving the generated discrete problems via the blended implementation of the methods.

Some linear algebra issues concerning the implementation of blended implicit methods

In this paper we discuss some linear algebra issues concerning the implementation of blended implicit methods (J. Comput. Appl. Math. 2000; 116:41–62, Appl. Numer. Math. 2002; 42:29–45, J. Comput. Appl. Math. 2004; 164–165:145–158, In Recent Trends in Numerical Analysis, Trigiante D (ed.), Nova Science Publication Inc.: New York, 2001; 81–105) for the numerical solution of ODEs. In particular, we describe the strategies, used in the numerical code BiM (J. Comput. Appl. Math. 2004; 164–165:145–158), for deciding whether re-evaluating the Jacobian and/or the factorization involved in the non-linear splitting for solving the discrete problem. Copyright

Matrix methods for radial Schrödinger eigenproblems defined on a semi-infinite domain

In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schrodinger 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 approximated 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.

A matrix method for fractional Sturm-Liouville problems on bounded domain

A matrix method for the solution of direct fractional Sturm-Liouville problems (SLPs) on bounded domains is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral method of orthogonal polynomials. The order of convergence of the eigenvalue approximations with respect to the matrix size is studied. Some numerical examples that confirm the theory and prove the competitiveness of the approach are finally presented.

Short-term recursions for fractional differential equations

This paper deals with the numerical solution of Fractional Differential Equations by means of m-step recursions. For the construction of such formulas, we study a technique based on a rational approximation of the generating functions of Fractional Backward Differentiation Formulas (FBDFs). The so-defined methods simulate very well the properties of the underlying FBDFs with important computational advantages. This fact becomes particularly evident especially in the case when they are used for solving problems arising from the semi-discretization of fractional partial differential equations.

Generalized Adams methods for fractional differential equations

We introduce a new family of fractional convolution quadratures based on generalized Adams methods for the numerical solution of fractional differential equations. We discuss their accuracy and linear stability properties. The boundary loci reported show that, when used as Boundary Value Methods, these schemes overcome the classical order barrier for A-stable methods.

P-stable boundary value methods for second order IVPs

We introduce a family of Linear Multistep Methods used as Boundary Value Methods for the numerical solution of initial value problems for second order ordinary differential equations of special type. The aim is to obtain P-stable methods with arbitrary order of accuracy. This result allows to overcome the order barrier established by Lambert and Watson which limited to p = 2 the maximum order of a P-stable Linear Multistep Method. In addition, an extension of the methods in the Exponential Fitting framework is also considered.

Shooting methods for a PT-symmetric periodic eigenvalue problem

We present a rigorous analysis of the performance of some one-step discretization schemes for a class of PT-symmetric singular boundary eigenvalue problem which encompasses a number of different problems whose investigation has been inspired by the 2003 article of Benilov et al. (J Fluid Mech 497:201–224, 2003). These discretization schemes are analyzed as initial value problems rather than as discrete boundary problems, since this is the setting which ties in most naturally with the formulation of the problem which one is forced to adopt due to the presence of an interior singularity. We also devise and analyze a variable step scheme for dealing with the singular points. Numerical results show better agreement between our results and those obtained from small-ϵ asymptotics than has been shown in results presented hitherto.