Jesús Vigo-Aguiar

Analysis of a Numerical Dynamic Programming Algorithm Applied to Economic Models

In this paper, the authors develop a discretized version of the dynamic programming algorithm and study its convergence and stability properties. They show that the computed value function converges quadratically to the true value function and that the computed value function converges linearly, as the mesh size of the discretization converges to zero; further, the algorithm is stable. The authors also discuss several aspects of the implementation of their procedures as applied to some commonly studied growth models.

Family of Twelve Steps Exponential Fitting Symmetric Multistep Methods for the Numerical Solution of the Schrödinger Equation

In the present paper we present a family of twelve steps symmetric multistep methods. The explicit part of new family of methods is applied to the scattering problems of the radial Schrödinger equation. This application shows the efficiency of the new family of methods.

Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrödinger equation

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems

In this paper a dissipative exponentially-fitted method for the numerical integration of the Schrodinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrodinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrodinger equation indicates the power of the new approach.

Weak Second Order Conditions for Stochastic Runge--Kutta Methods

A general procedure to construct weak methods for the numerical solution of stochastic differential systems is presented. As in the deterministic case, the procedure consists of comparing the stochastic expansion of the approximation with the corresponding Taylor scheme. In this way the authors obtain the order conditions that a stochastic Runge--Kutta method must satisfy to have weak order two. Explicit examples of generalizations of the classical family of second order two-stage explicit Runge--Kutta methods are shown. Also numerical examples are presented.

A General Procedure For the Adaptation of Multistep Algorithms to the Integration of Oscillatory Problems

This paper introduces a general technique for the construction of multistep methods capable of integrating, without local truncation error, homogeneous linear ODEs with constant coefficients, including those, in particular, that result in oscillatory solutions. Moreover, these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies, even confluent ones that occur from nonhomogeneous terms in the differential equation. Our procedure allows the derivation of many of the existing codes with similar properties, as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency. The properties of the methods are studied within a general framework, and numerical examples are presented. These demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes, particularly when integrating the class of equations with perturbed oscillatory solutions. The methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known, as occurs in satellite orbit propagation. The underlying ideas have already been applied to the improvement of some Chebyshev methods that are not multistep.

Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods

In this note, simplecticity conditions easy to handle for constructing symplectic Runge-Kutta-Nystrom methods fitted to trigonometric functions are given. These conditions generalize that of [1] when the frequencies tends to zero.

On the frequency choice in trigonometrically fitted methods

The choice of frequency in trigonometrically fitted methods is a fundamental question, especially if long-term prediction is considered. For linear oscillators, the frequency of the method is the same as the frequency of the solution of the differential equation. However, for nonlinear problems the frequency of the method is, in general, different from the frequency of the true solution. We present some experiments showing how the frequency depends strongly on certain values.

A Parallel Boundary Value Technique for Singularly Perturbed Two-Point Boundary Value Problems

A class of singularly perturbed two-point boundary-value problems (BVPs) for second-order ordinary differential equations (DEs) is considered here. In order to obtain numerical solution to these problems, an iterative non-overlapping domain decomposition method is suggested. The BVPs are independent in each subdomain and one can use parallel computers to solve these BVPs. One of the characteristics of the method is that the number of processors available is a free parameter of the method. Practical experiments on a Silicon Graphics Origin 200, with 4 MIPS R10000 processors have been performed, showing the reliability and performance of the proposed parallel schemes. Error estimates for the solution and numerical examples are provided.

A numerical algorithm for singular perturbation problems exhibiting weak boundary layers

Abstract A class of singularly perturbed two-point boundary-value problems (BVP) for second-order ordinary differential equations is considered here. To avoid the numerical difficulties in the solution to these problems, we divide the domain into two subdomains. The first BVP is a layer domain problem and the second BVP is a regular domain problem. Error estimates are derived for the numerical solution. Numerical examples are provided in support of the proposed method.