A preconditioned domain decomposition algorithm for the solution of the elliptic Neumann problem

A new preconditioned conjugate gradient (PCG)-based domain decomposition method is given for the solution of linear equations arising in the finite element method applied to the elliptic Neumann problem. The novelty of the proposed method is in the recommended preconditioner which is constructed by using cyclic matrix. The resulting preconditioned algorithms are well suited to parallel computation.     

An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem

Summary. This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem: , for , subject to , where . We assume that on , which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter . We present numerical results in support of this result. Received February 25, 1994     

Thermal degradation of two liquid fuels and detonation tests for pulse detonation engine studies

The use of liquid fuels such as kerosene is of interest for the pulse detonation engine (PDE). Within this context, the aim of this work, which is a preliminary study, was to show the feasibility to initiate a detonation in air with liquid-fuel pyrolysis products, using energies and dimensions of test facility similars to those of PDEs. Therefore, two liquids fuels have been compared, JP10, which is a synthesis fuel generally used in the field of missile applications, and decane, which is one of the major components of standard kerosenes (F-34, Jet A1, ...). The thermal degradation of these fuels was studied with two pyrolysis processes, a batch reactor and a flow reactor. The temperatures varied from 600°C to 1,000°C and residence times for the batch reactor and the flow reactor were, respectively, between 10–30 s and 0.1–2 s. Subsequently, the detonability of synthetic gaseous mixtures, which was a schematisation of the decomposition state after the pyrolysis process, has been studied. The detonability study, regarding nitrogen dilution and equivalence ratio, was investigated in a 50 mm-diameter, 2.5 m-long detonation tube. These dimensions are compatible with applications in the aircraft industry and, more particularly, in PDEs. Therefore, JP10 and decane were compared to choose the best candidate for liquid-fuel PDE studies. This paper was based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31 – August 5, 2005.     

A Hilbert Space Approach to Bounded Analytic Interpolation

We generalize several results on bounded analytic interpolation of Fitzgerald and Horn, which work by majorization by positive definite kernels, to the cases of several complex variables and operator-valued interpolation. Using a lemma of Kolmogorov, we complement a simplification due to Szafraniec in the proofs of the theorems. Received: November 21, 2006. Accepted: August 03, 2007.     

Existence of positive solutions for nonlinear three-point problems on time scales

In this paper, by using fixed point theorems in cones, we study the existence of at least one, two and three positive solutions of a nonlinear second-order three-point boundary value problem for dynamic equations on time scales. As an application, we also give some examples to demonstrate our results.     

Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution

Summary. We introduce linear semi-implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems. We study discretization of image selective smoothing equation of mean curvature flow type given by Alvarez, Lions and Morel ([3]). Solution of the level set equation of Osher and Sethian ([26], \[30]) is also included in the study. We prove and estimates for the proposed scheme and give existence of its (generalized) solution in every discrete time-scale step. Efficiency of the scheme is given by its linearity and stability. Preconditioned iterative solvers are used for computing arising linear systems. We present computational results related to image processing and plane curve evolution. Received April 25, 2000 / Revised version received June 11, 2001 / Published online November 15, 2001     

Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x ₀ and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated. Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003 This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.     

Runge–Kutta convolution quadrature methods for well-posed equations with memory

Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity, of the exact solution. Numerical illustrations are provided.