Relativistic properties and invariants of the Du Fort–Frankel scheme for the one-dimensional Schrödinger equation

Dellar, Paul J.

doi:10.1016/j.jcpx.2019.100004

Cited by 2 publications

(6 citation statements)

References 40 publications

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“…(2) Comparing with invariants-preserving Du Fort-Frankel-type FDMs for Schrödinger equations in [7,8,18,20,30,39], the current algorithms possess the following distinguishing features:…”

Section: Discussionmentioning

confidence: 99%

“…In [18], the development and convergence of a mass-and energy-preserving Du Fort-Frankeltype FDM, which has been generalized to numerical solutions of multi-dimensional Schrödinger equations on different geometries in [20], were derived for 1D Schrödinger equations in detail. As for more details, please refer to the references [7,8,18,20,29,30,39] and the related references therein. However, although those Du Fort-Frankel-type FDMs for Schrödinger equations can preserve the discrete mass and energy conservative laws, and provide reliable solutions, they are still conditionally stable, and suffer from strong grid restrictions.…”

Section: Introductionmentioning

confidence: 99%

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Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations With Wave Operator

Li¹

2024

JCM

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Du Fort-Frankel finite difference method (FDM) was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is an explicit and unconditionally von Neumann stable scheme. However, there has been no research work on numerical solutions of nonlinear Schrödinger equations with wave operator by using Du Fort-Frankel-type finite difference methods (FDMs). In this study, a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional (1D) and two-dimensional (2D) nonlinear Schrödinger equations with wave operator. By using the discrete energy method, it is shown that their solutions possess the discrete energy and mass conservative laws, and conditionally converge to exact solutions with an order of O(τ 2 + h 2x + (τ /hx) 2 ) for 1D problem and an order of O(τ 2 + h 2 x + h 2 y + (τ /hx) 2 + (τ /hy) 2 ) for 2D problem in H 1 -norm. Here, τ denotes time-step size, while, hx and hy represent spatial meshsizes in x-and y-directions, respectively. Then, by introducing a stabilized term, a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised. They not only preserve the discrete energies and masses, but also own much better stability than original schemes. Finally, numerical results demonstrate the theoretical analyses.

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“…(2) Comparing with invariants-preserving Du Fort-Frankel-type FDMs for Schrödinger equations in [7,8,18,20,30,39], the current algorithms possess the following distinguishing features:…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations With Wave Operator

Li¹

2024

JCM

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“…This now matches (14), except ∇•M has been replaced with ∇•M (0) = Θ∇B using the properties of the optimal TRT collision operator. We still have a kinetic equation for Λ, even though ( 34) is expressed solely in terms of u and B, because (34) contains two derivatives with respect to time.…”

mentioning

confidence: 96%

“…This finite difference scheme evolves the mass density across three time levels, and reduces to the Du Fort-Frankel scheme for the diffusion equation [17]. This derivation holds much more generally than the transformations connecting specific one-dimensional, two-velocity kinetic models and the Du Fort-Frankel scheme [1,10,14]. Conversely, Fučík & Straka [19] and Bellotti et al [3] have recently introduced algorithms to construct equivalent macroscopic finite difference schemes across multiple time levels for any lattice Boltzmann equation.…”

mentioning

confidence: 99%

“…The discrete equation (33) for evolving B only contains the equilibrium Λ (0) , an antisymmetric tensor, due to the properties of the optimal TRT collision operator. In general one would obtain a discrete evolution equation involving Λ, which is not antisymmetric according to (14). Equation ( 33) is thus a mimetic finite difference scheme [33,37,43] that exactly satisfies a discrete analog of the vector identity ∇•(∇×(u×B) ≡ 0.…”

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confidence: 99%

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A magic two-relaxation-time lattice Boltzmann algorithm for magnetohydrodynamics

Dellar

2023

DCDS-S

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The two-relaxation-time collision operator in discrete kinetic theory models collisions between particles by grouping them into pairs with antiparallel velocities. It prescribes a linear relaxation towards equilibrium with one rate for the even combination of distribution functions for each pair, and another rate for the odd combination. We reformulate this collision operator using relaxation rates for the forward-propagating and backward-propagating combinations instead. An optimal pair of relaxation rates sets the forwardpropagating combination of each pair of distributions to equilibrium. Only the backward-propagating non-equilibrium distributions remain. Applying this result twice gives closed discrete equations for evolving the macroscopic variables alone across three time levels. We split the equivalent equations into a firstorder system: a conservation law and a kinetic equation for the flux. All other quantities are evaluated at equilibrium. We apply this formalism to the magnetic field in a lattice Boltzmann scheme for magnetohydrodynamics. The antisymmetric part of the kinetic equation matches the Maxwell-Faraday equation and Ohm's law. The symmetric part matches the hyperbolic divergence cleaning model. The discrete divergence of the magnetic field remains zero, to within round-off error, when the initial magnetic field is the discrete curl of a vector potential. We have thus constructed a mimetic or constrained transport scheme for magnetohydrodynamics.

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Relativistic properties and invariants of the Du Fort–Frankel scheme for the one-dimensional Schrödinger equation

Cited by 2 publications

References 40 publications

Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations With Wave Operator

Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations With Wave Operator

A magic two-relaxation-time lattice Boltzmann algorithm for magnetohydrodynamics

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