Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations

Gao, Yali; Mei, Liquan

doi:10.1016/j.apnum.2016.06.003

Cited by 18 publications

(5 citation statements)

References 38 publications

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“…If the saturations S m g and S m w at time step m are given, we can implicitly solve ψ m+1 o by the pressure Equation (16), and update the saturations S m+1 w and S m+1 g by the saturation Equations (19) and (20). Consequently, the oil phase saturation S m+1 o can be calculated by 1 − S m+1 w − S m+1 g .…”

Section: Standard Impes Schemementioning

confidence: 99%

“…Numerical simulation for subsurface flows has been extensively applied in industry, such as in the management of groundwater energy and waste pollutants, petroleum engineering, and exploitation of geothermal energy [1][2][3][4][5][6][7][8][9][10][11][12][13]. The main numerical methods in the simulation include the Fully Implicit scheme (FI) [14][15][16][17] and the IMplicit EXplicit scheme (IMEX) [18][19][20]. In the FI scheme, all the unknowns can be derived implicitly.…”

Section: Introductionmentioning

confidence: 99%

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Improved IMPES Scheme for the Simulation of Incompressible Three-Phase Flows in Subsurface Porous Media

et al. 2021

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In this work, an improved IMplicit Pressure and Explicit Saturation (IMPES) scheme is proposed to solve the coupled partial differential equations to simulate the three-phase flows in subsurface porous media. This scheme is the first IMPES algorithm for the three-phase flow problem that is locally mass conservative for all phases. The key technique of this novel scheme relies on a new formulation of the discrete pressure equation. Different from the conventional scheme, the discrete pressure equation in this work is obtained by adding together the discrete conservation equations of all phases, thus ensuring the consistency of the pressure equation with the three saturation equations at the discrete level. This consistency is important, but unfortunately it is not satisfied in the conventional IMPES schemes. In this paper, we address and fix an undesired and well-known consequence of this inconsistency in the conventional IMPES in that the computed saturations are conservative only for two phases in three-phase flows, but not for all three phases. Compared with the standard IMPES scheme, the improved IMPES scheme has the following advantages: firstly, the mass conservation of all the phases is preserved both locally and globally; secondly, it is unbiased toward all phases, i.e., no reference phases need to be chosen; thirdly, the upwind scheme is applied to the saturation of all phases instead of only the referenced phases; fourthly, numerical stability is greatly improved because of phase-wise conservation and unbiased treatment. Numerical experiments are also carried out to demonstrate the strength of the improved IMPES scheme.

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Section: Standard Impes Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Improved IMPES Scheme for the Simulation of Incompressible Three-Phase Flows in Subsurface Porous Media

et al. 2021

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“…There exists a huge literature for the numerical solution of the NLSE, especially for 1D problems. Numerical methods for 2D NLSE include the alternating direction implicit (ADI) method [62], implicit-explicit multistep method [29], high order compact finite difference schemes [59], and local discontinuous Galerkin method [61], the implicit Crank-Nicolson finite difference (CNFD), semi-implicit finite difference (SIFD), time splitting spectral and pseudo spectral methods [6,7].…”

Section: Nonlinear Schrödinger Equationmentioning

confidence: 99%

Energy preserving model order reduction of the nonlinear Schrödinger equation

Karasözen

Uzunca

2018

Adv Comput Math

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An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.

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“…The

ψ false(x, t false)

describes fast time scale component of the electric field raised by electrons and the

ϕ false(x, t false)

represents reproduction of ion density from its equilibrium. The nonlinear term

{| ψ |}^{2} ψ

defines the nonlinear self‐interaction of electric field as in Schrödinger and Klein–Gordon equations [3, 25].

c

is the spreading speed of wave,

α

is a real‐valued constant and

λ

is a dimensionless constant [75].…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

Oruç

2022

Numerical Methods Partial

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This study presents numerical simulations of generalized two‐dimensional (2D) and three‐dimensional (3D) Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, which are coupled nonlinear partial differential equations. A meshless collocation method based on barycentric rational interpolation is developed for space variable of the KGZ equations. For time discretization, an explicit low storage fourth order Runge Kutta method is proposed after transforming KGZ equations to system of ordinary differential equations by introducing auxiliary variables. L∞ and L2 error norms for some test problems are computed. Obtained numerical results and comparisons with finite element methods indicate that barycentric rational interpolation method is an efficient method for solving multidimensional generalized KGZ system numerically.

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Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations

Cited by 18 publications

References 38 publications

Improved IMPES Scheme for the Simulation of Incompressible Three-Phase Flows in Subsurface Porous Media

Improved IMPES Scheme for the Simulation of Incompressible Three-Phase Flows in Subsurface Porous Media

Energy preserving model order reduction of the nonlinear Schrödinger equation

Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

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