A combined discontinuous Galerkin method for the dipolar Bose–Einstein condensation

Discrete Dynamics in Nature and Society

Shao

2017

Self Cite

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.

Section: The Temporal Discretizationmentioning

confidence: 99%

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

Discrete Dynamics in Nature and Society

Shao

2017

Self Cite

“…The similar method was extended in [7] to solve the compressible miscible displacement problem. The DGFE techniques have been applied by the authors of this paper [13,14,23,24], to nonlinear partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

A combined discontinuous Galerkin finite element method for miscible displacement problem

Zhu

Journal of Computational and Applied Mathematics

et al. 2017

Self Cite

A combined discontinuous Galerkin finite element method for miscible displacement problem, Journal of Computational and Applied Mathematics (2016), http://dx. Abstract.A new combined method is constructed for solving incompressible miscible displacement in porous media. In this procedure, a hybrid mixed element method is constructed for the pressure and velocity equation, while a symmetric discontinuous Galerkin finite element method is proposed for the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also conserves the local mass balance. The stability and consistency of the method are analyzed and the optimal error estimate in L ∞ (L 2 ) for velocity and concentration and the super convergence in L ∞ (H 1 ) for pressure are derived. Finally, some numerical results are presented.

“…There are many methods discretizing the normalized gradient flow in the imaginary time. These methods include spectral (pseudospectral) methods [3,5], finite difference method (FDM) [6,7], and discontinuous Galerkin (DG) method [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Compact Finite Difference Method for the Solution of the Gross-Pitaevskii Equation

Advances in Condensed Matter Physics

Liu

Zhao

2015

We present an efficient, unconditionally stable, and accurate numerical method for the solution of the Gross-Pitaevskii equation. We begin with an introduction on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem. Then we present a new numerical method, CFDM-AIF method, which combines compact finite difference method (CFDM) in space and array-representation integration factor (AIF) method in time. The key features of our methods are as follows: (i) the fourth-order accuracy in space andrth (r≥2) accuracy in time which can be achieved and (ii) the significant reduction of storage and CPU cost because of array-representation technique for efficient handling of exponential matrices. The CFDM-AIF method is implemented to investigate the ground and first excited state solutions of the Gross-Pitaevskii equation in two-dimensional (2D) and three-dimensional (3D) Bose-Einstein condensates (BECs). Numerical results are presented to demonstrate the validity, accuracy, and efficiency of the CFDM-AIF method.