2,054 research outputs found
On systems of differential equations with extrinsic oscillation
We present a numerical scheme for an efficient discretization of nonlinear systems of differential equations subjected to highly oscillatory perturbations. This method is superior
to standard ODE numerical solvers in the presence of high frequency forcing terms,and is based on asymptotic expansions of the solution in inverse powers of the oscillatory
parameter w, featuring modulated Fourier series in the expansion coefficients. Analysis of numerical stability and numerical examples are included
On second-order differential equations with highly oscillatory forcing terms
We present a method to compute efficiently solutions of systems of ordinary differential equations that possess highly oscillatory forcing terms. This approach is based on asymptotic expansions in inverse powers of the oscillatory parameter,and features two fundamental advantages with respect to standard ODE solvers: rstly, the construction of the numerical solution is more efficient when the system is highly oscillatory, and secondly, the cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided, motivated by problems in electronic engineering
Efficient computation of delay differential equations with highly oscillatory terms.
This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation
Differential equations with general highly oscillatory forcing terms
The concern of this paper is in expanding and computing initial-value problems of the form y' = f(y) + hw(t) where the function hw oscillates rapidly for w >> 1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators hw(t) = ÎŁm am(t)eim!t and they can be used as an organising principle for very accurate and aordable numerical solvers. However, there is no similar theory for more general oscillators and there are
sound reasons to believe that approximations of this kind are unsuitable in that setting. We follow in this paper an alternative route, demonstrating that, for a much more general family of oscillators, e.g. linear combinations of functions of the form ei!gk(t), it is possible to expand y(t) in a different manner. Each rth term in the expansion is for some & > 0 and it can be represented as an r-dimensional highly oscillatory integral. Since computation of multivariate highly oscillatory integrals is fairly well understood, this provides a powerful method for an effective discretisation of a numerical solution for a large family of highly oscillatory ordinary differential equations
Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions
For trigonometric and modified trigonometric integrators applied to
oscillatory Hamiltonian differential equations with one or several constant
high frequencies, near-conservation of the total and oscillatory energies are
shown over time scales that cover arbitrary negative powers of the step size.
This requires non-resonance conditions between the step size and the
frequencies, but in contrast to previous results the results do not require any
non-resonance conditions among the frequencies. The proof uses modulated
Fourier expansions with appropriately modified frequencies.Comment: 26 page
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