Blended General Linear Methods based on Boundary Value Methods in the GBDF family

Among the methods for solving ODE-IVPs, the class of General Linear Methods (GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome typical drawbacks of the previous classes of methods. For example, order barriers for stable LMF and the problem of order reduction for RK methods. Nevertheless, these goals are usually achieved at the price of a higher computational cost. Consequently, many efforts have been made 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. In particular, this approach has been recently developed, resulting in a new family of L-stable GLMs of arbitrarily high order, whose theory is here completed and fully worked-out. Moreover, for each one of such methods, it is possible to define a corresponding Blended GLM which is equivalent to it from the point of view of the stability and order properties. These blended methods, in turn, allow the definition of efficient nonlinear splittings for solving the generated discrete problems. A few numerical tests, confirming the excellent potential of such blended methods, are also reported.Comment: 22 pages, 8 figure

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

A matrix method for the solution of direct fractional Sturm-Liouville problems on bounded domain 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

A corrected spectral method for Sturm-Liouville problems with unbounded potential at one endpoint

In this paper, we shall derive a spectral matrix method for the approximation of the eigenvalues of (weakly) regular and singular Sturm-Liouville problems in normal form with an unbounded potential at the left endpoint. The method is obtained by using a Galerkin approach with an approximation of the eigenfunctions given by suitable combinations of Legendre polynomials. We will study the errors in the eigenvalue estimates for problems with unsmooth eigenfunctions in proximity of the left endpoint. The results of this analysis will be then used conveniently to determine low-cost and effective procedures for the computation of corrected numerical eigenvalues. Finally, we shall present and discuss the results of several numerical experiments which confirm the effectiveness of the approach.Comment: 28 pages, 5 figure

Matrix methods for radial Schr\"{o}dinger eigenproblems defined on a semi-infinite domain

In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schr\"{o}dinger 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

The BiM code for the numerical solution of ODEs

AbstractIn this paper we present the code BiM, based on blended implicit methods (J. Comput. Appl. Math. 116 (2000) 41; Appl. Numer. Math. 42 (2002) 29; Recent Trends in Numerical Analysis, Nova Science Publ. Inc., New York, 2001, pp. 81.), for the numerical solution of stiff initial value problems for ODEs. We describe in detail most of the implementation strategies used in the construction of the code, and report numerical tests comparing the code BiM with some of the best codes currently available. The numerical tests show that the new code compares well with existing ones. Moreover, the methods implemented in the code are characterized by a diagonal nonlinear splitting, which makes its extension for parallel computers very straightforward

On the construction and properties of m-step methods for FDEs

In this paper we consider the numerical solution of fractional differential equations by means of m-step recursions. The construction of such formulas can be obtained in many ways. Here we study a technique based on the rational approximation of the generating functions of fractional backward differentiation formulas (FBDFs). Accurate approximations lead to the definition of methods which simulate the underlying FBDF, with important computational advantages. Numerical experiments are presented

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