Numerical solution of nonlinear Schrödinger equation by using time‐space pseudo‐spectral method

Dehghan, Mehdi; Taleei, Ameneh

doi:10.1002/num.20468

Cited by 62 publications

(17 citation statements)

References 49 publications

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“…Of particular interest is GWRM CPU time and memory scaling with N t and N s . Using the case mentioned at the beginning of this section we have performed scans where ∈ [1,15] and ∈ [1,15] . It was found that CPU time scales as 1.0 1.43 and memory usage as 0.0 1.08 (for > 2).…”

Section: Accuracy -The Burger Equationmentioning

confidence: 99%

Optimizing Time-Spectral Solution of Initial-Value Problems

Scheffel¹,

Lindvall²

2018

AJCM

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Time-spectral solution of ordinary and partial differential equations is often regarded as an inefficient approach. The associated extension of the time domain, as compared to finite difference methods, is believed to result in uncomfortably many numerical operations and high memory requirements. It is shown in this work that performance is substantially enhanced by the introduction of algorithms for temporal and spatial subdomains in combination with sparse matrix methods. The accuracy and efficiency of the recently developed time spectral, generalized weighted residual method (GWRM) is compared to that of the explicit Lax-Wendroff method and the implicit Crank-Nicolson method. Three initial-value PDEs are employed as model problems; the 1D Burger equation, a forced 1D wave equation and a coupled system of 14 linearized ideal magnetohydrodynamic (MHD) equations. It is found that the GWRM is more efficient than the time-stepping methods at high accuracies. For timeaveraged solution of the two-time-scales, forced wave equation GWRM performance exceeds the finite difference methods by an order of magnitude both in terms of CPU time and memory requirement. Favourable scaling of CPU time and memory usage with the number of temporal and spatial subdomains is demonstrated for the MHD equations.

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Section: Accuracy -The Burger Equationmentioning

confidence: 99%

Optimizing Time-Spectral Solution of Initial-Value Problems

Scheffel¹,

Lindvall²

2018

AJCM

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“…More recently Dehghan and Taleei [19] found solutions to the non-linear Schrödinger equation, using a time-space pseudo-spectral method where the basis functions in time and space were constructed as a set of Lagrange interpolants.…”

Section: Generalized Weighted Residual Methodsmentioning

confidence: 99%

A time-spectral approach to numerical weather prediction

Scheffel

Lindvall

Yik

2018

Computer Physics Communications

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Finite difference methods are traditionally used for modelling the time domain in numerical weather prediction (NWP). Time-spectral solution is an attractive alternative for reasons of accuracy and efficiency and because time step limitations associated with causal, CFL-like critera are avoided. In this work, the Lorenz 1984 chaotic equations are solved using the time-spectral algorithm GWRM. Comparisons of accuracy and efficiency are carried out for both explicit and implicit time-stepping algorithms. It is found that the efficiency of the GWRM compares well with these methods, in particular at high accuracy. For perturbative scenarios, the GWRM was found to be as much as four times faster than the finite difference methods. A primary reason is that the GWRM time intervals typically are two orders of magnitude larger than those of the finite difference methods. The GWRM has the additional advantage to produce analytical solutions in the form of Chebyshev series expansions. The results are encouraging for pursuing further studies, including spatial dependence, of the relevance of time-spectral methods for NWP modelling.

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“…(4), were solved in the numerical form by means of a pseudospectral method, which was adjusted to the present model, following the lines of works [35] (the application of this method to soliton solutions of equations of the NLS type was recently elaborated in Ref. [36]). The integration domain of variable s, of width T ¼ 40, was sufficient to completely study the shape and dynamics of individual solitons, including those which feature extended ''tails" (simulations of interactions between solitons might require using a larger domain).…”

Section: The Numerical Methodsmentioning

confidence: 99%

Single- and multi-peak solitons in two-component models of metamaterials and photonic crystals

Chen

Malomed

2010

Optics Communications

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a b s t r a c tWe report results of the study of solitons in a system of two nonlinear-Schrödinger (NLS) equations coupled by the XPM interaction, which models the co-propagation of two waves in metamaterials (MMs). The same model applies to photonic crystals (PCs), as well as to ordinary optical fibers, close to the zero-dispersion point. A peculiarity of the system is a small positive or negative value of the relative group-velocity dispersion (GVD) coefficient in one equation, assuming that the dispersion is anomalous in the other. In contrast to earlier studied systems of nonlinearly coupled NLS equations with equal GVD coefficients, which generate only simple single-peak solitons, the present model gives rise to families of solitons with complex shapes, which feature extended oscillatory tails and/or a double-peak structure at the center. Regions of existence are identified for single-and double-peak bimodal solitons, demonstrating a broad bistability in the system. Behind the existence border, they degenerate into single-component solutions. Direct simulations demonstrate stability of the solitons in the entire existence regions. Effects of the group-velocity mismatch (GVM) and optical loss are considered too. It is demonstrated that the solitons can be stabilized against the GVM by means of the respective ''management" scheme. Under the action of the loss, complex shapes of the solitons degenerate into simple ones, but periodic compensation of the loss supports the complexity.

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Numerical solution of nonlinear Schrödinger equation by using time‐space pseudo‐spectral method

Cited by 62 publications

References 49 publications

Optimizing Time-Spectral Solution of Initial-Value Problems

Optimizing Time-Spectral Solution of Initial-Value Problems

A time-spectral approach to numerical weather prediction

Single- and multi-peak solitons in two-component models of metamaterials and photonic crystals

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