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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