A spectral method for solving nonhomogeneous Neumann boundary value problems on quadrilaterals

Wang, Tianjun; Sun, Tao

doi:10.1016/j.apnum.2020.05.025

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…The most popular DM for solving linear elliptic problems stated in the domain of simple (rectangular) geometry is the Fast Fourier Transform (FFT) algorithm. After its development in the second half of the 20th century [7,8], it has been applied ubiquitously in almost every scientific discipline because of its accuracy and cost-effectiveness [9][10][11][12][13][14][15][16][17][18]. Moreover, there are many commercially available packages that implement the FFT algorithm.…”

Section: Introductionmentioning

confidence: 99%

Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method

et al. 2023

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In this paper, we consider the 3D problem of laser-induced semiconductor plasma generation under the action of the optical pulse, which is governed by the set of coupled time-dependent non-linear PDEs involving the Poisson equation with Neumann boundary conditions. The main feature of this problem is the non-linear feedback between the Poisson equation with respect to induced electric field potential and the reaction-diffusion-convection-type equation with respect to free electron concentration and accounting for electron mobility (convection’s term). Herein, we focus on the choice of the numerical method for the Poisson equation solution with inhomogeneous Neumann boundary conditions. Despite the ubiquitous application of such a direct method as the Fast Fourier Transform for solving an elliptic problem in simple spatial domains, we demonstrate that applying a direct method for solving the problem under consideration results in a solution distortion. The reason for the Neumann problem’s solvability condition violation is the computational error’s accumulation. In contrast, applying an iterative method allows us to provide finite-difference scheme conservativeness, asymptotic stability, and high computation accuracy. For the iteration technique, we apply both an implicit alternating direction method and a new three-stage iteration process. The presented computer simulation results confirm the advantages of using iterative methods.

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Section: Introductionmentioning

confidence: 99%

Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method

et al. 2023

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A multi-domain spectral-Galerkin method for the Neumann problem on quadrilaterals

Ma,

Wang,

Shang

2024

Computers & Mathematics with Applications

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Multi-Stages Iterative Process for Conservative Economic Finite-Difference Schemes Realization for the Problem of Nonlinear Laser Pulse Interaction with a Medium

Trofimov

Loginova

Egorenkov

2021

Nonlinear Phenomena in Complex Systems

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We consider a problem of laser pulse interaction with a nonlinear medium which is accompanied by different nonlinear phenomena. Among them, we highlight the laser pulse self-action, optical bistability realization, formation of laser-induced complicated spatio-temporal structures. For computer modeling of these strongly nonlinear effects, using robust conservative numerical methods is required. Well-known, there are two widely applied approaches for the construction of numerical method: the conservative finite-difference schemes and additive finite-difference schemes (the split-step methods or decomposition methods). The first ones are non-economic, as a rule, while the second type of the methods is economic ones, however they possess well-known disadvantages. In our study, we joint advantages of both approaches by developing an original multi-stage iterative process for the conservative finite-difference scheme realization. Using computer simulation results, we demonstrate the feasibility of the proposed approach for investigating certain nonlinear optical phenomena.

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A spectral method for solving nonhomogeneous Neumann boundary value problems on quadrilaterals

Cited by 3 publications

References 18 publications

Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method

Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method

A multi-domain spectral-Galerkin method for the Neumann problem on quadrilaterals

Multi-Stages Iterative Process for Conservative Economic Finite-Difference Schemes Realization for the Problem of Nonlinear Laser Pulse Interaction with a Medium

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