Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

2014 ◽  
Vol 6 (4) ◽  
pp. 494-514 ◽  
Author(s):  
Yaming Chen ◽  
Songhe Song ◽  
Huajun Zhu
Keyword(s):  
Dirac Equation ◽  
Hamiltonian System ◽  
Numerical Experiments ◽  
Pseudospectral Method ◽  
Splitting Methods ◽  
Fourier Pseudospectral Method ◽  
Composition Method ◽  
Nonlinear Dirac Equation ◽  
Symplectic Scheme ◽  
Symplectic System

AbstractIn this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

Structure-Preserving Wavelet Algorithms for the Nonlinear Dirac Model

2017 ◽  
Vol 9 (4) ◽  
pp. 964-989 ◽  
Author(s):  
Xu Qian ◽  
Hao Fu ◽  
Songhe Song
Keyword(s):  
Dirac Equation ◽  
Theoretical Analysis ◽  
Conservation Laws ◽  
Numerical Experiments ◽  
Quantum Physics ◽  
Structure Preserving ◽  
Splitting Technique ◽  
Nonlinear Dirac Equation

AbstractThe nonlinear Dirac equation is an important model in quantum physics with a set of conservation laws and a multi-symplectic formulation. In this paper, we propose energy-preserving and multi-symplectic wavelet algorithms for this model. Meanwhile, we evidently improve the efficiency of these algorithms in computations via splitting technique and explicit strategy. Numerical experiments are conducted during long-term simulations to show the excellent performances of the proposed algorithms and verify our theoretical analysis.

Nearly perfectly matched layer absorber for viscoelastic wave equations

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. T335-T345
Author(s):  
Enjiang Wang ◽  
José M. Carcione ◽  
Jing Ba ◽  
Mamdoh Alajmi ◽  
Ayman N. Qadrouh
Keyword(s):  
Wave Equations ◽  
Pseudospectral Method ◽  
Perfectly Matched Layer ◽  
Zener Model ◽  
Fourier Pseudospectral Method ◽  
Time Stepping ◽  
First Case ◽  
Stress Strain Relation ◽  
Viscoelastic Wave ◽  
Spatial Derivatives

We have applied the nearly perfectly matched layer (N-PML) absorber to the viscoelastic wave equation based on the Kelvin-Voigt and Zener constitutive equations. In the first case, the stress-strain relation has the advantage of not requiring additional physical field (memory) variables, whereas the Zener model is more adapted to describe the behavior of rocks subject to wave propagation in the whole frequency range. In both cases, eight N-PML artificial memory variables are required in the absorbing strips. The modeling simulates 2D waves by using two different approaches to compute the spatial derivatives, generating different artifacts from the boundaries, namely, 16th-order finite differences, where reflections from the boundaries are expected, and the staggered Fourier pseudospectral method, where wraparound occurs. The time stepping in both cases is a staggered second-order finite-difference scheme. Numerical experiments demonstrate that the N-PML has a similar performance as in the lossless case. Comparisons with other approaches (S-PML and C-PML) are carried out for several models, which indicate the advantages and drawbacks of the N-PML absorber in the anelastic case.

scholarly journals On the nonlinear Dirac equation on noncompact metric graphs

2021 ◽  
Vol 278 ◽  
pp. 326-357
Author(s):  
William Borrelli ◽  
Raffaele Carlone ◽  
Lorenzo Tentarelli
Keyword(s):  
Dirac Equation ◽  
Metric Graphs ◽  
Nonlinear Dirac Equation

scholarly journals Solitary waves in the nonlinear Dirac equation in the presence of external driving forces

2016 ◽  
Vol 49 (6) ◽  
pp. 065402 ◽  
Author(s):  
Franz G Mertens ◽  
Fred Cooper ◽  
Niurka R Quintero ◽  
Sihong Shao ◽  
Avinash Khare ◽  
...  
Keyword(s):  
Dirac Equation ◽  
Solitary Waves ◽  
Driving Forces ◽  
Nonlinear Dirac Equation ◽  
External Driving

scholarly journals Prediction of Dynamic Stability Using Mapped Chebyshev Pseudospectral Method

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jae-Young Choi ◽  
Dong Kyun Im ◽  
Jangho Park ◽  
Seongim Choi
Keyword(s):  
Unsteady Flow ◽  
Three Dimensional ◽  
Pseudospectral Method ◽  
Accurate Method ◽  
Spectral Approach ◽  
Fourier Pseudospectral Method ◽  
Chebyshev Pseudospectral Method ◽  
Fourier Pseudospectral ◽  
Chebyshev Pseudospectral ◽  
Good Agreement

A mapped Chebyshev pseudospectral method is extended to solve three-dimensional unsteady flow problems. As the classical Chebyshev spectral approach can lead to numerical instabilities due to ill conditioning of the spectral matrix, the Chebyshev points are evenly redistributed over the domain by an inverse sine mapping function. The mapped Chebyshev pseudospectral method can be used as an alternative time-spectral approach that uses a Chebyshev collocation operator to approximate the time derivative terms in the unsteady flow governing equations, and the method can make general applications to both nonperiodic and periodic problems. In this study, the mapped Chebyshev pseudospectral method is employed to solve three-dimensional periodic problem to verify the spectral accuracy and computational efficiency with those of the Fourier pseudospectral method and the time-accurate method. The results show a good agreement with both of the Fourier pseudospectral method and the time-accurate method. The flow solutions also demonstrate a good agreement with the experimental data. Similar to the Fourier pseudospectral method, the mapped Chebyshev pseudospectral method approximates the unsteady flow solutions with a precise accuracy at a considerably effective computational cost compared to the conventional time-accurate method.

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