Mass- and Energy-Conserved Numerical Schemes for Nonlinear Schrödinger Equations

Feng, Xin; Liu, Hailiang; Ma, Shu

doi:10.4208/cicp.2019.js60.05

Cited by 21 publications

(6 citation statements)

References 21 publications

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“…Due to a complexity of the partial differential equations, such as GLE, and nonlinear Schrödinger equation (NLSE), Gross-Pitaevskii equation, Klein-Gordon equation, and Korteweg-De Vries equation, and others, the exact solutions can be obtained only in particular cases or maybe with strict assumptions. Therefore, solving these problems via computer simulations is an attractive issue and many numerical methods have been proposed and investigated [42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61].…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, many authors have directed their efforts towards developing conservative FDSs for a solution of the nonlinear problem. Among them we emphasize the problem of finding numerical solutions of the NLSE [49][50][51][52][53][54][55][56][57][58][59][60][61]. In [51], a splitting method for the cubic NLSE on a torus that possesses a long-time near-conservation of energy is applied and investigated.…”

Section: Introductionmentioning

confidence: 99%

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Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

et al. 2022

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In this study, our attention is focused on deriving integrals of motion (conservation laws; invariants) for the problem of an optical pulse propagation in an optical fiber containing an optical amplifier or attenuator because, to date, such invariants are absent in the literature. The knowledge of a problem’s invariants allows us develop finite-difference schemes possessing the conservativeness property, which is crucial for solving nonlinear problems. Laser pulse propagation is governed by the nonlinear Ginzburg–Landau equation. Firstly, the problem’s conservation laws are developed for the various parameters’ relations: for a linear case, for a nonlinear case without considering the linear absorption, and for a nonlinear case accounting for the linear absorption and homogeneous shift of the pulse’s phase. Hereafter, the Crank–Nicolson-type scheme is constructed for the problem difference approximation. To demonstrate the conservativeness of the constructed implicit finite-difference scheme in the sense of preserving difference analogs of the problem’s invariants, the corresponding theorems are formulated and proved. The problem of the finite-difference scheme’s nonlinearity is solved by means of an iterative process. Finally, several numerical examples are presented to support the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

et al. 2022

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“…There have been lots of numerical methods in the market to solve the TDKS equation in the time domain, people may refer to [3,7,14] and references therein for detail. People may also refer to [11,16,28] for numerical methods of Schrödinger equation. Among those grid-based numerical methods, the finite difference methods [1], the finite element methods [3,8,9,17,18,27], the discontinuous Galerkin methods [20], the wavelet methods [12] etc.…”

Section: Introductionmentioning

confidence: 99%

An Implicit Solver for the Time-Dependent Kohn-Sham Equation

Yang¹

2021

NMTMA

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The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation. However, the efficiency issue prevents the practical applications of those implicit methods. In this paper, an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation. The efficiency issue is partially resolved by three approaches, i.e., an h-adaptive mesh method is proposed to effectively restrain the size of the discretized problem, a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization, as well as the OpenMP based parallelization of the algorithm. The numerical convergence, the ability on preserving the physical properties, and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.

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“…Both schemes preserve mass and energy conservations at the discrete level. More recently, Feng et al [13] constructed a class of second-order mass-and energy-conserving schemes for the NLS equation, including the modified Crank-Nicolson method, the implicit leapfrog method and a class of modified backward differentiation formulae as special cases.…”

Section: Introductionmentioning

confidence: 99%

High-order Mass- and Energy-conserving SAV-Gauss Collocation Finite Element Methods for the Nonlinear Schrödinger Equation

Feng

2021

SIAM J. Numer. Anal.

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A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form O(h p + τ k+1 ) in the L ∞ (0, T ; H 1 )-norm is established, where h and τ denote the spatial and temporal mesh sizes, respectively, and (p, k) is the degree of the space-time finite elements. Numerical experiments are provided to validate the theoretical results on the convergence rates and conservation properties. The effectiveness of the proposed methods in preserving the shape of a soliton wave is also demonstrated by numerical results.

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Mass- and Energy-Conserved Numerical Schemes for Nonlinear Schrödinger Equations

Cited by 21 publications

References 21 publications

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

An Implicit Solver for the Time-Dependent Kohn-Sham Equation

High-order Mass- and Energy-conserving SAV-Gauss Collocation Finite Element Methods for the Nonlinear Schrödinger Equation

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