Francesca Mazzia

Solving ordinary differential equations by generalized Adams methods: properties and implementation techniques☆

Generalized Adams methods of order 3, 5, 7 and 9 are used to find numerical solutions of initial value problems. The effectiveness of these methods for the treatment of stiff problems is shown on the basis of their attractive properties and an efficient technique to deal with the algebraic nonlinear systems representing the discrete counterpart of the continuous problem. Numerical examples are also presented in which an experimental code based on these methods is compared with two well known codes for ODEs. The numerical results are quite satisfactory and suggest that these methods may have a useful role in the solution of stiff ODEs.

Block-Boundary Value Methods for the Solution of Ordinary Differential Equations

Block-boundary value methods are considered and applied to solve initial value problems. Results on their convergence and stability properties are presented and their intermediate position between the class of multistep and Runge--Kutta methods is established.

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.

B-Spline Linear Multistep Methods and their Continuous Extensions

In this paper, starting from a sequence of results which can be traced back to I. J. Schoenberg, we analyze a class of spline collocation methods for the numerical solution of ordinary differential equations (ODEs) with collocation points coinciding with the knots. Such collocation methods are naturally associated to a special class of linear multistep methods, here called B-spline (BS) methods, which are able to generate the spline values at the knots. We prove that, provided the additional conditions are appropriately chosen, such methods are all convergent and

Numerical approximation of nonlinear BVPs by means of BVMs

A

Boundary value methods based on Adams-type methods

-stable. The convergence property of the BS methods is naturally inherited by the related spline extensions, which, by the way, are easily and safely computable using their B-spline representation.

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

Boundary Value Methods (BVMs) would seem to be suitable candidates for the solution of nonlinear Boundary Value Problems (BVPs). They have been successfully used for solving linear BVPs together with a mesh selection strategy based on the conditioning of the linear systems. Our aim is to extend this approach so as to use them for the numerical approximation of nonlinear problems. For this reason, we consider the quasi-linearization technique that is an application of the Newton method to the nonlinear differential equation. Consequently, each iteration requires the solution of a linear BVP In order to guarantee the convergence to the solution of the continuous nonlinear problem, it is necessary to determine how accurately the linear BVPs must be solved. For this goal, suitable stopping criteria on the residual and on the error for each linear BVP are given. Numerical experiments on stiff problems give rather satisfactory results, showing that the experimental code, called TOM, that uses a class of BVMs and the quasi-linearization technique, may be competitive with well known solvers for BVPs.

Convergence and Stability of Multistep Methods Solving Nonlinear Initial Value Problems

Abstract The aim of this paper is to derive Boundary Value Methods (BVMs) based on k-step Adams-type methods for the solution of initial value problems. BVMs lead to a discrete boundary value problem which needs one initial and k − 1 final conditions. We prove that the choice of boundary conditions, instead of the usual initial conditions, improves the stability properties of the classical Adams methods. For example, methods of order up to 6 are almost BV-A-stable, and those of order up to 9 are BV-A0-stable.

Parallel block preconditioning for the solution of boundary value methods

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.