An algorithm for the computation of consistent initial values for differential–algebraic equations

We use boundary value methods to compute consistent initial values for fully implicit nonlinear differential-algebraic equations. The obtained algorithm uses variable order formulae and a deferred correction technique to evaluate the error. A rigorous theory is stated for nonlinear index 1, 2 and 3 DAEs of Hessenberg form. Numerical tests on classical index 1, 2 and 3 DAE problems are reported. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multistep Methods for Conservative Problems

We discuss the use of linear multistep methods for the solution of conservative (in particular Hamiltonian) problems. Despite the lack of good results concerning their behaviour, linear multistep methods have been satisfactorily used by researchers in applicative fields. We try to elucidate the requirements that a linear multistep method should fulfill in order to be suitable for the integration of such problems, relating our analysis to their use as boundary value methods. We collect a number of results obtained for the linear autonomous case and also consider some numerical tests on the Kepler problem.Work supported by GNCS and MIUR.     

A Hybrid Mesh Selection Strategy Based on Conditioning for Boundary Value ODE Problems

An appropriate mesh selection strategy is one of the fundamental tools in designing robust codes for differential problems, especially if the codes are required to work for difficult multi scale problems. Most of the existing codes base the mesh selection on an estimate of the error (or the residual). Our strategy, based on the estimation of two parameters characterising the conditioning of the continuous problem, as well as on an estimate of the error, not only permits us to obtain a well adapted mesh, thus reducing the cost of the code, but also provides a measure of the conditioning of both continuous and discrete problems.     

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

A boundary value approach to the numerical solution of initial value problems by multistep methos †

A boundary value appraoch to the numerical solution of initial value problems by means of linear multistep methods is presented. This theory is based on the study of linear difference equations when their general solution is computed by imposing boundary conditions. All the main stability and convergence properties of the obtained methods are investigated abd compared to those of the classical multistep methods. Then, as an example, new itegration formulas, called extended trapezoidal rules, are derived. For any order they have the same stability properties (in the sense of the definitions given in this paper) of the trapezoidal rule, which is the first method in this class. Some numerical examples are presented to confirm the theoretical expectations and to allow us to trust a future code based on boundary value methods.     

Generalization of Backward Differentiation Formulas for Parallel Computers

We analyze an extension of backward differentiation formulas, used as boundary value methods, that generates a class of methods with nice stability and convergence properties. These methods are obtained starting from the boundary value GBDFs class, and are in the class of EBDF-type methods. We discuss different ways of using these linear multistep formulas in order to have efficient parallel implementations. Numerical experiments show their effectiveness.     

Product Gauss quadrature rules vs. cubature rules in the meshless local Petrov–Galerkin method

A crucial point in the implementation of meshless methods such as the meshless local Petrov–Galerkin (MLPG) method is the evaluation of the domain integrals arising over circles in the discrete local weak form of the governing partial differential equation. In this paper we make a comparison between the product Gauss numerical quadrature rules, which are very popular in the MLPG literature, with cubature formulas specifically constructed for the approximation of an integral over the unit disk, but not yet applied in the MLPG method, namely the spherical, the circularly symmetrical and the symmetric cubature formulas. The same accuracy obtained with 64×64 points in the product Gauss rules may be obtained with symmetric quadrature formulas with very few points.     

A Test Set for stiff Initial Value Problem Solvers in the open source software R: Package deTestSet

In this paper we present the R package deTestSet that includes challenging test problems written as ordinary differential equations (ODEs), differential algebraic equations (DAEs) of index up to 3 and implicit differential equations (IDEs). In addition it includes 6 new codes to solve initial value problems (IVPs). The R package is derived from the Test Set for Initial Value Problem Solvers available at http://www.dm.uniba.it/~testset which includes documentation of the test problems, experimental results from a number of proven solvers, and Fortran subroutines providing a common interface to the defining problem functions. Many of these facilities are now available in the R package deTestSet, which comprises an R interface to the test problems and to most of the Fortran solvers. The package deTestSet is free software which is distributed under the GNU General Public License, as part of the R open source software project.     

Eigenvalues and Quasi-Eigenvalues of Banded Toeplitz Matrices: Some Properties and Applications

In this paper we consider the spectrum and quasi-eigenvalues of a family of banded Toeplitz matrices and define their extensions to the generalized eigenvalue problem. A diagonal similarity transformation on such matrices that allows a suitable modification of the region containing the quasi-eigenvalues is reported. Two kind of applications have been analyzed: the computation of the eigenvalues and the asymptotic spectra of Toeplitz matrices and the solution of block banded quasi-Toeplitz linear systems that arise after the discretization of an ODE using a boundary value method.