The numerical study of a nonlinear one-dimensional Dirac equation

Alvárez, A.; Kuo, P. K.; Vasquez, Luis

doi:10.1016/0096-3003(83)90027-9

Cited by 32 publications

(26 citation statements)

References 3 publications

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“…Guo et al [12] consider spectral and pseudospectral semidiscrete approximations of the solution to the Cauchy problem for (1.2), proving a local-time discrete L 2 error estimate (cf. [3]). Shao and Tang [25] discretize (1.2) using a discontinuous finite element method in space and an explicit Runge-Kutta method in time.…”

Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

“…[7,25,27]) and on the other hand the iterations needed to solve nonlinear systems of algebraic equations which is the outcome of an implicit method (cf. [3,7,8,13,15]). …”

Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

“…3.11) used in other works on the analysis of numerical methods for problem (1.2) or (1.1) in order to show convergence in a discrete L 2 norm (see, e.g., [7,8]). Also, the error analysis presented in [3,12] assumes that the final time T is small enough. Finally, in Section 4, we explain implementation issues for our finite difference method, and we show results from numerical experiments that confirm the order of convergence of the method.…”

Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

“…In particular, Alvarez and Carreras [2] consider the physical problem (1.2) and investigate, via numerical computations, the interaction dynamics for solitary waves. Alvarez et al [3] formulate the numerical method used in [2], which combines a second order finite diffence discretization in space with a Crank-Nicolson time stepping, and provide a local-time error estimate in the discrete L 2 norm. A complete error analysis in the discrete L 2 norm for the Crank-Nicolson finite difference method and in the case of periodic boundary conditions is given by De Frutos [7], who also proves the discrete L 2 convergence of an explicit leap-frog finite difference method for (1.2) with periodic boundary conditions and of an implicit box-method for (1.2) with a special type of boundary conditions.…”

Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

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

Bournaveas

Zouraris

2012

ESAIM: M2AN

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Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

“…[7,25,27]) and on the other hand the iterations needed to solve nonlinear systems of algebraic equations which is the outcome of an implicit method (cf. [3,7,8,13,15]). …”

Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

Section: An Implicit-explicit Finite Difference Methodsmentioning

confidence: 99%

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

Bournaveas

Zouraris

2012

ESAIM: M2AN

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“…Therefore, numerical techniques become important to solve the NLD equation and analyze its properties. From the numerical point of view, Alvarez et al proposed a secondorder Crank-Nicholson (CN) scheme [11] and the linearized CN scheme [12]. In 1989, split-step spectral schemes were presented by de Frutos and Sanz-Serna [13].…”

Section: Introductionmentioning

confidence: 99%

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

Zhang

Shao

2017

Discrete Dynamics in Nature and Society

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A high-order accuracy numerical method is proposed to solve the (1 + 1)-dimensional nonlinear Dirac equation in this work. We construct the compact finite difference scheme for the spatial discretization and obtain a nonlinear ordinary differential system. For the temporal discretization, the implicit integration factor method is applied to deal with the nonlinear system. We therefore develop two implicit integration factor numerical schemes with full discretization, one of which can achieve fourth-order accuracy in both space and time. Numerical results are given to validate the accuracy of these schemes and to study the interaction dynamics of the nonlinear Dirac solitary waves.

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Time‐splitting methods with charge conservation for the nonlinear Dirac equation

Li¹,

Li²,

Shi

2017

Numerical Methods Partial

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In this work, four numerical time‐splitting methods are proposed for the (1 + 1)‐dimensional nonlinear Dirac equation. All of these methods (or schemes) are proved to satisfy the charge conservation in the discrete level. To enhance the computation efficiency, the block Thomas algorithm is adopted. Numerical experiments are given to test the accuracy order for these schemes, to simulate numerically the binary collision including two standing waves and two moving solitons, meanwhile, the dynamic properties for the nonlinear Dirac equation are discussed. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1582–1602, 2017

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The numerical study of a nonlinear one-dimensional Dirac equation

Cited by 32 publications

References 3 publications

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

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

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

Time‐splitting methods with charge conservation for the nonlinear Dirac equation

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