T.E. Simos2 Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia and Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, GR-221 00 Tripolis, Greece This is to remind you that the affiliation of the Journal of Numerical Analysis, Industrial and Applied Mathematics is: Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM) This is very important
Sunday, January 30. 2022
F. Iavernaro3
Dipartimento di Matematica, Universita di Bari, I-70125 Bari, Italy
D. Trigiante
Dipartimento di Energetica, Universita di Firenze, I-50134 Firenze, Italy
Received 11 March, 2009; accepted in revised form 23 April, 2009
Dedicated to John Butcher on the occasion of his 75th birthday
Abstract: We define a class of arbitrary high order symmetric one-step methods that, when applied to Hamiltonian systems, are capable of precisely conserving the Hamiltonian function when this is a polynomial, whatever the initial condition and the stepsize h used.
The key idea to devise such methods is the use of the so called discrete line integral, the discrete counterpart of the line integral in conservative vector fields. This approach naturally suggests a formulation of such methods in terms of block Boundary Value Methods, although they can be recast as Runge-Kutta methods, if preferred.
Thursday, December 9. 2010
Keywords: Two-point Boundary Value Problems, singular perturbation problems, finite difference schemes, upwind method, mesh variation.
Abstract: We propose a simple and quite efficient code to solve singular perturbation problems when the perturbation parameter ǫ is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses highly variable stepsize to fit the boundary regions with relatively few points. An extensive numerical test section shows the effectiveness of the proposed technique on linear problems.
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Thursday, December 9. 2010
Keywords: Linear differential systems, time window, spectral approximation, waveform relaxation.
Abstract: We establish a relation between the length T of the integration window of a linear differential equation x′+Ax = b and a spectral parameter s∗. This parameter is determined by comparing the exact solution x(T) at the end of the integration window to the solution of a linear system obtained from the Laplace transform of the differential equation by freezing the system matrix. We propose a method to integrate the relation s∗ = s∗(T) into the determination of the interval of rapid convergence of waveform relaxation iterations. The method is illustrated with a few numerical examples.
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Thursday, December 9. 2010
Keywords: Numerical methods for ordinary differential equations, General Linear Methods, Boundary Value Methods (BVMs), Generalized Backward Differentiation Formulae (GBDF), Blended Implicit Methods, blended iteration.
Abstract: 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 [11]. In particular, in [8], 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.
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Thursday, December 9. 2010
Keywords: seismic tomography, spectral-element method, adjoint-method, Australia
Abstract: We propose a novel technique for seismic waveform tomography on continental scales. This is based on the fully numerical simulation of wave propagation in complex Earth models, the inversion of complete waveforms and the quantification of the waveform discrepancies through a specially designed phase misfit. The numerical solution of the equations of motion allows us to overcome the limitations of ray theory and of finite normal mode summations. Thus, we can expect the tomographic models to be more realistic and physically consistent. Moreover, inverting entire waveforms reduces the non-uniqueness of the tomographic problem. Following the theoretical descriptions of the forward and inverse problem solutions, we present preliminary results for the upper mantle structure in the Australasian region.
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Thursday, December 9. 2010
Keywords: Stiff problems, A-stability, stability barriers, order stars, order arrows, nonlinear stability.
Abstract: We discuss two events with profound implications on the way initial value problems are solved numerically. The first was the identification of stiffness as a widely spread phenomenon affecting the ability to obtain useful results. The second was the definition of A-stability as an important approach to overcoming the effect of stiffness. Not only was the idea associated with A-stability significant in its own time but it has had long term effects including new theoretical questions as well as the tools for solving them.
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Tuesday, November 2. 2010
Date of Online Publication: 02/11/2010 Keywords: Edge detection, discontinuities, wavelets, polyharmonic splines.
Abstract: In this paper we consider the problem of detecting, from a finite discrete set of points, the curves across which a two-dimensional function is discontinuous. We propose a strategy based on wavelets which allows to discriminate the edge points from points in which the function has steep gradients or extrema.
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Tuesday, November 2. 2010
Date of Online Publication: 02/11/2010 Keywords: Partial Differential Equations, Meshless Method, Radial Basis Function, Radial Point Interpolation
Abstract: A meshless method based on radial point interpolation was recently developed as an effective tool for solving partial differential equations, and has been widely applied to a number of different problems. In addition to the primary advantage of the meshless methods that the computation is performed without any connectivity information between field nodes, the radial point interpolation-based meshless method has several advantages such as the stability of the shape functions and simple implementation of boundary condition enforcement. This paper introduces a new scheme for the radial point interpolation-based meshless method. This method enables fast computation by modifying the construction and evaluation of the shape functions. Numerical examples are also presented to show that a reliable solution can be obtained with low computational cost.
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Tuesday, November 2. 2010
Date of Online Publication: 02/11/2010 Keywords: Existence results, beam vibration, damping, hybrid model.
Abstract: We consider the solvability of a hybrid model for the vibration of a vertical slender structure mounted on an elastic seating. The slender structure is modeled as a Rayleigh beam and gravity is taken into account. The seating and foundation block are modeled as rigid bodies connected by elastic springs with damping mechanisms. We show how an existence result for a general linear vibration problem in variational form may be applied to the weak variational problem for this system.
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Tuesday, November 2. 2010
Date of Online Publication: 02/11/2010 Keywords: Stochastic Differential Equations, Additive Noise, Numerical Solution, Runge– Kutta methods Periodic orbits, Numerical drift. Authors: Foivos Xanthos and George Papageorgiou
Abstract: In this paper we study the numerical treatment of Stochastic Differential Equations with additive noise and one dimensional Wiener process. We develop two, three and four stage Runge–Kutta methods which attain deterministic order up to four and stochastic order up to one and a half specially constructed for this class of problems. Numerical tests and comparisons with other known methods in the solution of various problems justify our effort, especially for our three stages methods.
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Tuesday, November 2. 2010
Date of Online Publication: 02/11/2010 Keywords: Time reversal symmetry, Reversible Hamiltonian systems, Symmetric methods, Periodic orbits, Numerical drift. Authors: Pages:
Abstract: When approximating reversible Hamiltonian problems, the presence of a “drift” in the numerical values of the Hamiltonian is sometimes experienced, even when reversible methods of integration are used. In this paper we analyze the phenomenon by using a more precise definition of time reversal symmetry for both the continuous and the discrete problems. A few examples are also presented to support the analysis.
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