Efficient SAV approach for imaginary time gradient flows with applications to one- and multi-component Bose-Einstein Condensates

Zhuang, Qingqu; Shen, Jie

doi:10.1016/j.jcp.2019.06.043

Cited by 41 publications

(13 citation statements)

References 21 publications

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“…In the last two decades, there have been a plethora of numerical methods developed to compute ground states of BECs, including normalized gradient flow methods based on the Hamiltonian (the energy) of the GP equation [3,6,7,12,13,16,18,21,27,30,47,66,71,73], and methods for the nonlinear eigenvalue problem (see e.g., [23,24,28,35,67,72] and references therein) based on time-independent GP equation as well as constrained optimization techniques [8, 15, 22, 32-34, 64, 69]. The normalized gradient flow strategy is considered from the PDE perspective, leading to numerical algorithms for a dissipative system.…”

Section: Introductionmentioning

confidence: 99%

Accelerated numerical algorithms for steady states of Gross-Pitaevskii equations coupled with microwaves

Wang¹,

Wang²

2022

Preprint

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We present two accelerated numerical algorithms for single-component and binary Gross-Pitaevskii (GP) equations coupled with microwaves (electromagnetic fields) in steady state.One is based on a normalized gradient flow formulation, called the ASGF method, while the other on a perturbed, projected conjugate gradient approach for the nonlinear constrained optimization, called the PPNCG method. The coupled GP equations are nonlocal in space, describing pseudospinor Bose-Einstein condensates (BECs) interacting with an electromagnetic field. Our interest in this study is to develop efficient, iterative numerical methods for steady symmetric and central vortex states of the nonlocal GP equation systems. In the algorithms, the GP equations are discretized by a Legendre-Galerkin spectral method in a polar coordinate in two-dimensional (2D) space. The new algorithms are shown to outperform the existing ones through a host of benchmark examples, among which the PPNCG method performs the best. Additional numerical simulations of the central vortex states are provided to demonstrate the usefulness and efficiency of the new algorithms.

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Section: Introductionmentioning

confidence: 99%

Accelerated numerical algorithms for steady states of Gross-Pitaevskii equations coupled with microwaves

Wang¹,

Wang²

2022

Preprint

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“…Due to its attractive properties, the SAV approach has been intensively studied in these years. It has been applied to many PDEs, for example, the two-dimensional sine-Gordon equation [6], the fractional nonlinear Schrödinger equation [12], the Camassa-Holm equation [21], and the imaginary time gradient flow [42]. Shen and Xu [32] and Li, Shen and Rui [22] conducted a convergence analysis of SAV schemes.…”

Section: Introductionmentioning

confidence: 99%

Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy

Kemmochi¹,

Sato²

2021

Preprint

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In this paper, we present a novel investigation of the so-called SAV approach, which is a framework to construct linearly implicit geometric numerical integrators for partial differential equations with variational structure. SAV approach was originally proposed for the gradient flows that have lower-bounded nonlinear potentials such as the Allen-Cahn and Cahn-Hilliard equations, and this assumption on the energy was essential. In this paper, we propose a novel approach to address gradient flows with unbounded energy such as the KdV equation by a decomposition of energy functionals. Further, we will show that the equation of the SAV approach, which is a system of equations with scalar auxiliary variables, is expressed as another gradient system that inherits the variational structure of the original system. This expression allows us to construct novel higher-order integrators by a certain class of Runge-Kutta methods. We will propose second and fourth order schemes for conservative systems in our framework and present several numerical examples.

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“…Thanks to their simplicity, efficiency, accuracy and generality, the SAV and GSAV approaches have received much attention recently, they, along with their various variations/extensions, have been used to construct unconditionally energy stable schemes for a large class of nonlinear systems, including various gradient flows (see, for instance, [5,6,27,23,33,40,39]), gradient flows with other global constraints (see, for instance, [8]), Navier-Stokes equations and related systems (see, for instance, [25,22,15,24]), time fractional PDEs [17], conservative or Hamiltonian systems (see, for instance, [3,42,2,14]), and many more. However, in the original SAV approach and its various variants, the discrete value of the SAV is not directly linked to its definition at the continuous level, and this may lead to a loss of accuracy when the time step is not sufficiently small.…”

mentioning

confidence: 99%

“…may converge to a value different from 1, leading to a wrong steady state solution (see Table 1 in [42]). One possible remedy for this is to monitor the ratio…”

mentioning

confidence: 99%

A generalized SAV approach with relaxation for dissipative systems

Zhang,

Shen

2022

Preprint

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The scalar auxiliary variable (SAV) approach [30] and its generalized version GSAV proposed in [19] are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccurate solutions if the time step is not sufficiently small. Inspired by the relaxed SAV method proposed in [20] for gradient flows, we propose in this paper a generalized SAV approach with relaxation (R-GSAV) for general dissipative systems. The R-GSAV approach preserves all the advantages of the GSAV appraoch, in addition, it dissipates a modified energy that is directly linked to the original free energy. We prove that the k-th order implicit-explicit (IMEX) schemes based on R-GSAV are unconditionally energy stable, and we carry out a rigorous error analysis for k = 1, 2, 3, 4, 5. We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach.

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Efficient SAV approach for imaginary time gradient flows with applications to one- and multi-component Bose-Einstein Condensates

Cited by 41 publications

References 21 publications

Accelerated numerical algorithms for steady states of Gross-Pitaevskii equations coupled with microwaves

Accelerated numerical algorithms for steady states of Gross-Pitaevskii equations coupled with microwaves

Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy

A generalized SAV approach with relaxation for dissipative systems

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