Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Bournaveas, Nikolaos; Zouraris, Georgios E.

doi:10.1051/m2an/2011071

Cited by 18 publications

(10 citation statements)

References 25 publications

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“…which is obtained by approximating explicitly the nonlinear terms but implicitly the linear terms and will be called by SI. It is worth noting that such semi-implicit scheme has been studied for the NLD equation with the quadric scalar self-interaction in [46].…”

Section: Several Finite Difference Schemesmentioning

confidence: 99%

“…Note in passing that the two-humped profile was first pointed out by Shao and Tang [49] and later gotten noticed by other researchers [16]. Besides the often-used CN [48] and RKDG methods [50], there exist many other numerical schemes for solving the (1+1)-dimensional NLD equation: split-step spectral schemes [52], the linearized CN scheme [53], the semi-implicit scheme [54] [46], Legendre rational spectral methods [55], multi-symplectic Runge-Kutta methods [56], adaptive mesh methods [57] etc. The fourth-order accurate RKDG method [50] is very appropriate for investigating the interaction dynamics of the NLD solitary waves due to their ability to capture the discontinuous or strong gradients without pro-ducing spurious oscillations, and thus performs better than the second-order accurate CN scheme [48].…”

Section: Introductionmentioning

confidence: 99%

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Numerical methods for nonlinear Dirac equation

Jian

Shao

Tang

2013

Journal of Computational Physics

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This paper presents a review of the current state-of-the-art of numerical methods for nonlinear Dirac (NLD) equation. Several methods are extendedly proposed for the (1+1)-dimensional NLD equation with the scalar and vector self-interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge, energy and linear momentum. Those methods are the Crank-Nicolson (CN) schemes, the linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme, a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes. The nonlinear subproblems resulted from the OS schemes are analytically solved by fully exploiting the local conservation laws of the NLD equation. The effectiveness of the various numerical methods, with special focus on the error growth and the computational cost, is illustrated on two numerical experiments, compared to two high-order accurate Runge-Kutta discontinuous Galerkin methods. Theoretical and numerical comparisons show that the high-order accurate OS schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency. A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics of the NLD solitary waves under the scalar and vector self-interaction. The results show that the interaction dynamics of two NLD solitary waves depend on the exponent power of the self-interaction in the NLD equation; collapse happens after collision of two equal one-humped NLD solitary waves under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.Comment: 39 pages, 13 figure

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Section: Several Finite Difference Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical methods for nonlinear Dirac equation

Jian

Shao

Tang

2013

Journal of Computational Physics

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“…Following some examples displayed in [8], we intend to test the scheme (2.7) against exact solutions of a nonlinear Dirac equation (compare with (2.1), no time-potential):…”

Section: Well-balanced Upwind Schemementioning

confidence: 99%

A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

Gosse

2014

Bit Numer Math

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The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the L 2 norm despite being explicit in time. This construction is wellsuited for particles for which the reference velocity is of the order of c, the speed of light. Moreover, when c diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a "diffusive limit" emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that c∆ x → 0). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed.

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“…In particular, we have (βΦ ε ,Φ ε ) = |φ ε 1 | 2 − |φ ε 2 | 2 . The nonlinear Dirac equation (1.1) has been widely considered in the literature [6,4,30,24] as a reduced mathematical model. It was originally derived in [25] as a four-component equations in 3D for describing the spinor field with nonlinear coupling.…”

Section: Introductionmentioning

confidence: 99%

“…With a fixed ε > 0, the well-posedness of the Cauchy problem (1.1) has been studied and established. We refer to [4,21,23,24,3] and the references therein for detailed analytical results. Various numerical methods including finite difference time domain (FDTD) methods and operator splitting methods have also been considered in [26,29,30,4,15] for solving the nonlinear Dirac equation in the classical regime, i.e.…”

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confidence: 99%

Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime

Lemou

Méhats

Zhao

2017

Communications in Mathematical Sciences

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Abstract. We apply the two-scale formulation approach to propose uniformly accurate (UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic limit regime. The nonlinear Dirac equation involves two small scales ε and ε 2 with ε → 0 in the nonrelativistic limit regime. The small parameter causes high oscillations in time which brings severe numerical burden for classical numerical methods. We transform our original problem as a two-scale formulation and present a general strategy to tackle a class of highly oscillatory problems involving the two small scales ε and ε 2 . Suitable initial data for the two-scale formulation is derived to bound the time derivatives of the augmented solution. Numerical schemes with uniform (with respect to ε ∈ (0,1]) spectral accuracy in space and uniform first order or second order accuracy in time are proposed. Numerical experiments are done to confirm the UA property.

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Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Cited by 18 publications

References 25 publications

Numerical methods for nonlinear Dirac equation

Numerical methods for nonlinear Dirac equation

A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime

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