Study of instability of the Fourier split-step method for the massive Gross–Neveu model

Lakoba, Taras I.

doi:10.1016/j.jcp.2019.109100

Cited by 3 publications

(2 citation statements)

References 47 publications

(193 reference statements)

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“…Numerical schemes for (61) have attracted considerable attention in the last few years; see, for example, [40,41] and references therein. We will simulate two of its exact soutions.…”

Section: Numerical Verificationmentioning

confidence: 99%

“…However, we will not test their performance relative to those of the MoC‐(N)pRK schemes due to, again, space limitation, as well as in order to maintain our focus on the latter schemes. It should be noted that the fourth‐order method proposed in [40] is (unlike the MoC‐(p)RK4) restricted to using periodic BC; it was demonstrated in [41] that certain solutions (62) exhibit numerical instability for periodic (but not for nonreflecting) BC.…”

Section: Numerical Verificationmentioning

confidence: 99%

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Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

Lakoba

Jewell

2021

Numerical Methods Partial

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We develop method of characteristics schemes based on explicit Runge-Kutta and pseudo-Runge-Kutta third-and fourth-order solvers along the characteristics. Schemes based on Runge-Kutta solvers are found to be strongly unstable for certain physics-motivated models. In contrast, schemes based on pseudo-Runge-Kutta solvers are shown to be only weakly unstable for periodic boundary conditions and essentially stable for the more physically relevant nonreflecting boundary conditions. Our implementation of nonreflecting boundary conditions does not rely on interpolation. KEYWORDS coupled-mode equations, higher-order methods, method of characteristics 1 Here y ± and f ± are vectors of respective lengths N ± and subscripts denote partial differentiation. Functions f ± are, in general, nonlinear. Moreover, they, in principle, may contain small diffusion-like terms y ± xx. We will briefly comment on the latter possibility in the concluding section of this work, but in the main part of it we will assume that system (1a) is nondissipative.

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“…Numerical schemes for (61) have attracted considerable attention in the last few years; see, for example, [40,41] and references therein. We will simulate two of its exact soutions.…”

Section: Numerical Verificationmentioning

confidence: 99%

Section: Numerical Verificationmentioning

confidence: 99%

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

Lakoba

Jewell

2021

Numerical Methods Partial

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Stability of nonlinear Dirac solitons under the action of external potential

Mellado-Alcedo,

Quintero

2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The instabilities observed in direct numerical simulations of the Gross–Neveu equation under linear and harmonic potentials are studied. The Lakoba algorithm, based on the method of characteristics, is performed to numerically obtain the two spinor components. We identify non-conservation of energy and charge in simulations with instabilities, and we find that all studied solitons are numerically stable, except the low-frequency solitons oscillating in the harmonic potential over long periods of time. These instabilities, as in the case of the Gross–Neveu equation without potential, can be removed by imposing absorbing boundary conditions. The dynamics of the soliton is in perfect agreement with the prediction obtained using an Ansatz with only two collective coordinates, namely, the position and momentum of the center of mass. We employ the temporal variation of both field energy and momentum to determine the evolution equations satisfied by the collective coordinates. By applying the same methodology, we also demonstrate the spurious character of the reported instabilities in the Alexeeva–Barashenkov–Saxena model under external potentials.

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Direct expansion of Fourier extrapolator for one-way wave equation using Chebyshev polynomials of the second kind

2022

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The Fourier method for one-way wave propagation is efficient, but potentially inaccurate in complex media. The implicit finite-difference method can handle arbitrarily complex media, but can be inefficient in 3D and has limited dip bandwidth. We proposed a new Fourier method based on Chebyshev expansion of the second kind. Both theoretical analyses and numerical experiments show that the proposed method is comprehensively superior to a similar method based on Chebyshev expansion of the first kind in terms of balanced amplitude and error tolerance. Within the dip bandwidth from 0 to 65°, the fourth-order form of our method has an error tolerance of 2%, which is about one-third that of Chebyshev expansion of the first kind. Our method is also superior to the implicit finite-difference method in several important aspects: effective bandwidth, computational efficiency, numerical dispersion and two-way splitting error. It can be easily extended from 2D to 3D compared with the finite-difference method and from low orders to high orders compared with the optimized Chebyshev-Fourier method. The proposed method shows better imaging results of the SEG/EAGE model by providing a well-focused salt dome, flank and bottom as well as the detailed structures beneath the salt body, compared with the implicit finite-difference method and Chebyshev expansion of the first kind; meanwhile, our method has less imaging artifacts since it can better position the reflectors.

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Study of instability of the Fourier split-step method for the massive Gross–Neveu model

Cited by 3 publications

References 47 publications

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

Stability of nonlinear Dirac solitons under the action of external potential

Direct expansion of Fourier extrapolator for one-way wave equation using Chebyshev polynomials of the second kind

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