A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon

Boyd, John P.

doi:10.1007/978-3-642-15337-2_10

Cited by 7 publications

(5 citation statements)

References 42 publications

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“…Due to the fact that the proposed method starts exhibiting numerical instability for a relatively low number of basis functions , the true convergence rate cannot be truly recognized form Figure 3. Although that, from observing this figure, we get the impression that the method has an exponential convergence rate, from our previous work with the exponential basis [14] and analysis of convergence rates for series approximations for functions with singularities [42], we expect that the convergence speed is lower and closer to root exponential.…”

Section: Resultsmentioning

confidence: 84%

Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

Jovanović

Kais

Alharbi

2014

Journal of Applied Mathematics

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We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work.

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Section: Resultsmentioning

confidence: 84%

Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

Jovanović

Kais

Alharbi

2014

Journal of Applied Mathematics

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“…(A.71) is more accurately described as the "Euler acceleration" (or "Abel-Euler summability"[9]) of Eq. (A.70); see also Ref [8]. and those cited therein.…”

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confidence: 83%

Finite-amplitude acoustics under the classical theory of particle-laden flows

Jordan¹

2019

Evolution Equations &Amp; Control Theory

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We consider acoustic propagation in a particle-laden fluid, specifically, a perfect gas, under a model system based on the theories of Marble (1970) and Thompson (1972). Our primary aim is to understand, via analytical methods, the impact of the particle phase on the acoustic velocity field. Working under the finite-amplitude approximation, we investigate singular surface and traveling wave phenomena, as admitted by both phases of the flow. We show, among other things, that the particle velocity field admits a singular surface one order higher than that of the gas phase, that the particleto-gas density ratio plays a number of critical roles, and that traveling wave solutions are only possible for sufficiently small values of the Mach number. i.e., the particle specific heat (at constant pressure) is negligibly small; see Thompson [34, pp. 553-556]. The immediate (practical) consequence of this assumption is τ T = 0 (⇒ the absence of thermal inertia), where τ T ∝ c pp is the thermal relaxation time.

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“…It does not add significant computational costs. Unfortunately, the same does not hold for the adaptive filters (Boyd 2011;Tadmor and Tanner 2005;Tanner 2006) because, if we vary coefficients depending on position, we can no longer use the substitution of (2.20) which leads to the COS method.…”

Section: Filtering and The Cos Methodsmentioning

confidence: 99%

On the Application of Spectral Filters in a Fourier Option Pricing Technique

Ruijter¹,

Versteegh

Oosterlee

2013

SSRN Journal

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When Fourier techniques are applied to specific problems from computational finance with nonsmooth functions, the so-called Gibbs phenomenon may become apparent. This seriously affects the efficiency and accuracy of the numerical results. For example, the variance gamma asset price process gives rise to algebraically decaying Fourier coefficients, resulting in a slowly converging Fourier series. We apply spectral filters to achieve faster convergence. Filtering is carried out in Fourier space; the series coefficients are pre-multiplied by a decreasing function. This does not add any significant computational costs. Tests with different filters show how the algebraic index of convergence is improved.

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A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon

Cited by 7 publications

References 42 publications

Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

Finite-amplitude acoustics under the classical theory of particle-laden flows

On the Application of Spectral Filters in a Fourier Option Pricing Technique

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