12,295 research outputs found
A family of Continuous Third Derivative Block Methods for solving stiff systems of first order ordinary differential equations
AbstractThis paper presents a family of Continuous Third Derivative Block Methods (CTDBM) of order k+3 for the solution of stiff systems of ordinary differential equations. The approach uses the collocation and interpolation technique to generate the main Continuous Third Derivative method (CTDM) which is then used to obtain the additional methods that are combined as a single block methods. Analysis of the methods show that the method is L-stable up to order eight. Numerical examples are given to illustrate the accuracy and efficiency of the proposed method
On the resolution power of Fourier extensions for oscillatory functions
Functions that are smooth but non-periodic on a certain interval possess
Fourier series that lack uniform convergence and suffer from the Gibbs
phenomenon. However, they can be represented accurately by a Fourier series
that is periodic on a larger interval. This is commonly called a Fourier
extension. When constructed in a particular manner, Fourier extensions share
many of the same features of a standard Fourier series. In particular, one can
compute Fourier extensions which converge spectrally fast whenever the function
is smooth, and exponentially fast if the function is analytic, much the same as
the Fourier series of a smooth/analytic and periodic function.
With this in mind, the purpose of this paper is to describe, analyze and
explain the observation that Fourier extensions, much like classical Fourier
series, also have excellent resolution properties for representing oscillatory
functions. The resolution power, or required number of degrees of freedom per
wavelength, depends on a user-controlled parameter and, as we show, it varies
between 2 and \pi. The former value is optimal and is achieved by classical
Fourier series for periodic functions, for example. The latter value is the
resolution power of algebraic polynomial approximations. Thus, Fourier
extensions with an appropriate choice of parameter are eminently suitable for
problems with moderate to high degrees of oscillation.Comment: Revised versio
Computer-Aided Derivation of Multi-scale Models: A Rewriting Framework
We introduce a framework for computer-aided derivation of multi-scale models.
It relies on a combination of an asymptotic method used in the field of partial
differential equations with term rewriting techniques coming from computer
science.
In our approach, a multi-scale model derivation is characterized by the
features taken into account in the asymptotic analysis. Its formulation
consists in a derivation of a reference model associated to an elementary
nominal model, and in a set of transformations to apply to this proof until it
takes into account the wanted features. In addition to the reference model
proof, the framework includes first order rewriting principles designed for
asymptotic model derivations, and second order rewriting principles dedicated
to transformations of model derivations. We apply the method to generate a
family of homogenized models for second order elliptic equations with periodic
coefficients that could be posed in multi-dimensional domains, with possibly
multi-domains and/or thin domains.Comment: 26 page
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