An integrating factor for nonlinear Dirac equations

Hoz, Francisco de la; Vadillo, Fernando

doi:10.1016/j.cpc.2010.03.004

Cited by 13 publications

(15 citation statements)

References 11 publications

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“…For instance, in comparison with the table errors offered in [15], our results are more precise and, above all, the elapsed CPU time is much smaller. Bearing in mind the good behavior of our simulation for the (1 + 1)-dimensional problem and the fact that Matlab© also implements in a convenient way higher-dimensional discrete Fourier transforms and their inverses (in two variables, we have for instance fft2 and ifft2), we realized that small modifications in our codes would allow us to simulate successfully the NLD equation in two and three spatial dimensions, which is also done in [8].…”

Section: The Nonlinear Dirac Equationmentioning

confidence: 96%

“…where P and Q , which depend on the number of spatial variables, are given in [8]. In the Fourier space, the linear term does not appear, so we get the system of ODEs…”

Section: The Nonlinear Dirac Equationmentioning

confidence: 99%

“…In [8], we have presented a new numerical method to simulate the evolution in time of the nonlinear Dirac equation, in one, two and three spatial dimensions. More precisely, this method uses an integrating factor (see for example [1] or [5]) to remove the linear term in the Fourier space; then, the resulting system of ODEs is integrated in time by means of a fourthorder Runge-Kutta scheme.…”

Section: The Nonlinear Dirac Equationmentioning

confidence: 99%

“…Following the presentation of paper [15], we have considered in [8] several collisions; as, for example, the collision of three Dirac solitary waves:…”

Section: The Nonlinear Dirac Equationmentioning

confidence: 99%

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Numerical simulations of time-dependent partial differential equations

Hoz

Vadillo

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tWhen a time-dependent partial differential equation (PDE) is discretized in space with a spectral approximation, the result is a coupled system of ordinary differential equations (ODEs) in time. This is the notion of the method of lines (MOL), and the resulting set of ODEs is stiff; the stiffness may be even exacerbated sometimes. The linear terms are the primarily responsible for the stiffness, with a rapid exponential decay of some modes (as in a dissipative PDE), or a rapid oscillation of some modes (as in a dispersive PDE). Therefore, for a time-dependent PDE which combines low-order nonlinear terms with higher-order linear terms, it is desirable to use a higher-order approximation both in space and in time.Along our research, we have focused on a particular case of spectral methods, the socalled pseudo-spectral methods, to solve numerically time-dependent PDEs using different techniques: an integrating factor, in de la Hoz and Vadillo (2010); an exponential time differencing method, in de la Hoz and Vadillo (2008); and differentiation matrices in the theoretical frame of matrix differential equations, in de la Vadillo (2012, 2013a,b). This paper, which is a unified review of those contributions, aims at providing a better understanding of those methods, by illustrating their variety and, more importantly, their power. Furthermore, we also give emphasis to choosing adequate schemes to advance in time.

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Section: The Nonlinear Dirac Equationmentioning

confidence: 96%

“…where P and Q , which depend on the number of spatial variables, are given in [8]. In the Fourier space, the linear term does not appear, so we get the system of ODEs…”

Section: The Nonlinear Dirac Equationmentioning

confidence: 99%

Section: The Nonlinear Dirac Equationmentioning

confidence: 99%

“…Following the presentation of paper [15], we have considered in [8] several collisions; as, for example, the collision of three Dirac solitary waves:…”

Section: The Nonlinear Dirac Equationmentioning

confidence: 99%

See 2 more Smart Citations

Numerical simulations of time-dependent partial differential equations

Hoz

Vadillo

2016

Journal of Computational and Applied Mathematics

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“…The CN finite difference method and the exponential wave integrator Fourier pseudospectral method were developed by Bao et al in 2016 [16]. Besides, there are some other numerical techniques, such as multisymplectic Runge-Kutta methods [17], an efficient adaptive mesh redistribution method [18], an integrating factor method [19], and time-splitting methods with charge conservation [20]. Some experts also discussed the spin-orbitcoupled Bose-Einstein condensates [21] and the MaxwellDirac system [22][23][24] which are related to the NLD equation.…”

Section: Introductionmentioning

confidence: 99%

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

Zhang

Shao

2017

Discrete Dynamics in Nature and Society

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A high-order accuracy numerical method is proposed to solve the (1 + 1)-dimensional nonlinear Dirac equation in this work. We construct the compact finite difference scheme for the spatial discretization and obtain a nonlinear ordinary differential system. For the temporal discretization, the implicit integration factor method is applied to deal with the nonlinear system. We therefore develop two implicit integration factor numerical schemes with full discretization, one of which can achieve fourth-order accuracy in both space and time. Numerical results are given to validate the accuracy of these schemes and to study the interaction dynamics of the nonlinear Dirac solitary waves.

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Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime

Ma,

Chen

2024

Numerical Methods Partial

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We present exponential wave integrator Fourier pseudospectral (EWI‐FP) methods and establish their error estimates of the fully discrete schemes for the Dirac equation in the massless and nonrelativistic regime. This regime involves a small dimensionless parameter where , and is inversely proportional to the speed of light. The solution exhibits highly oscillatory behavior in time and rapid wave propagation in space in this regime. Specifically, the time oscillations have a wavelength of , while the spatial oscillations have a wavelength of , with a wave speed of . We employ (symmetric) exponential wave integrators for temporal derivatives and Fourier spectral discretization for spatial derivatives. We rigorously derive the error bounds which explicitly depend on the mesh size , the time step and the small dimensionless parameter . The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (‐scalability) necessitates setting and when . Finally, some numerical examples are provided to validate the error bounds.

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An integrating factor for nonlinear Dirac equations

Cited by 13 publications

References 11 publications

Numerical simulations of time-dependent partial differential equations

Numerical simulations of time-dependent partial differential equations

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime

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