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
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 i...
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MDPI AG
2023-06-01
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| Series: | Mathematics |
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| author | Vyacheslav Trofimov Maria Loginova Vladimir Egorenkov Yongqiang Yang Zhongwei Yan |
| author_facet | Vyacheslav Trofimov Maria Loginova Vladimir Egorenkov Yongqiang Yang Zhongwei Yan |
| author_sort | Vyacheslav Trofimov |
| collection | DOAJ |
| description | 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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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T03:02:03Z |
| publishDate | 2023-06-01 |
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| spelling | doaj.art-d3cdc017943443f9ba5c685548bab4d12023-11-18T08:13:48ZengMDPI AGMathematics2227-73902023-06-011111256710.3390/math11112567Violation 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 MethodVyacheslav Trofimov0Maria Loginova1Vladimir Egorenkov2Yongqiang Yang3Zhongwei Yan4School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, ChinaThe Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaThe Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaSchool of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, ChinaSchool of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, ChinaIn 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.https://www.mdpi.com/2227-7390/11/11/25673D partial differential equationsNeumann problem solvabilityconservative finite-difference schemeiteration methoddirect methodinhomogeneous boundary conditions |
| spellingShingle | Vyacheslav Trofimov Maria Loginova Vladimir Egorenkov Yongqiang Yang Zhongwei Yan 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 Mathematics 3D partial differential equations Neumann problem solvability conservative finite-difference scheme iteration method direct method inhomogeneous boundary conditions |
| title | 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 |
| title_full | 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 |
| title_fullStr | 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 |
| title_full_unstemmed | 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 |
| title_short | 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 |
| title_sort | 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 |
| topic | 3D partial differential equations Neumann problem solvability conservative finite-difference scheme iteration method direct method inhomogeneous boundary conditions |
| url | https://www.mdpi.com/2227-7390/11/11/2567 |
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