Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method
This study investigates the numerical modeling of two-dimensional anisotropic diffusion processes involving a spatially localized and temporally limited energy or thermal source. The governing model is formulated as a parabolic partial differential equation, discretized in space using the Finite Ele...
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| Format: | Article |
| Language: | Indonesian |
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Universitas Jember
2025-06-01
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| Series: | Berkala Sainstek |
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| Online Access: | https://bst.jurnal.unej.ac.id/index.php/BST/article/view/53711 |
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| author | M. Ziaul Arif Millatuz Zahroh Sailah Ar Rizka |
| author_facet | M. Ziaul Arif Millatuz Zahroh Sailah Ar Rizka |
| author_sort | M. Ziaul Arif |
| collection | DOAJ |
| description | This study investigates the numerical modeling of two-dimensional anisotropic diffusion processes involving a spatially localized and temporally limited energy or thermal source. The governing model is formulated as a parabolic partial differential equation, discretized in space using the Finite Element Method (FEM) with linear triangular elements, and in time using both explicit and implicit Euler integration schemes. To ensure spatial accuracy, a dense mesh configuration is employed, which has been shown to produce smooth and representative solution distributions. Simulation results demonstrate that the implicit Euler method exhibits superior numerical stability across various time step sizes, whereas the explicit method requires significantly smaller time steps to remain stable. Analysis of the transient regime reveals that the numerical solution gradually converges toward a steady-state configuration once the source is deactivated. These findings confirm that the combination of FEM with implicit time integration and dense meshing is effective in capturing the spatiotemporal dynamics of anisotropic diffusion processes with localized sources, a phenomenon relevant to thermal analysis, anisotropic materials, and environmental modeling. |
| format | Article |
| id | doaj-art-f4255dc87617455194aa6ba88a83ff2b |
| institution | OA Journals |
| issn | 2339-0069 |
| language | Indonesian |
| publishDate | 2025-06-01 |
| publisher | Universitas Jember |
| record_format | Article |
| series | Berkala Sainstek |
| spelling | doaj-art-f4255dc87617455194aa6ba88a83ff2b2025-08-20T02:10:24ZindUniversitas JemberBerkala Sainstek2339-00692025-06-0113211913010.19184/bst.v13i2.5371165561Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element MethodM. Ziaul Arif0Millatuz Zahroh1Sailah Ar Rizka2Jurusan Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Jember, Jl. Kalimantan No. 37 Kampus Tegalboto, Jember, Jawa Timur, 68121, IndonesiaJurusan Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Jember, Jl. Kalimantan No. 37 Kampus Tegalboto, Jember, Jawa Timur, 68121, IndonesiaJurusan Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Jember, Jl. Kalimantan No. 37 Kampus Tegalboto, Jember, Jawa Timur, 68121, IndonesiaThis study investigates the numerical modeling of two-dimensional anisotropic diffusion processes involving a spatially localized and temporally limited energy or thermal source. The governing model is formulated as a parabolic partial differential equation, discretized in space using the Finite Element Method (FEM) with linear triangular elements, and in time using both explicit and implicit Euler integration schemes. To ensure spatial accuracy, a dense mesh configuration is employed, which has been shown to produce smooth and representative solution distributions. Simulation results demonstrate that the implicit Euler method exhibits superior numerical stability across various time step sizes, whereas the explicit method requires significantly smaller time steps to remain stable. Analysis of the transient regime reveals that the numerical solution gradually converges toward a steady-state configuration once the source is deactivated. These findings confirm that the combination of FEM with implicit time integration and dense meshing is effective in capturing the spatiotemporal dynamics of anisotropic diffusion processes with localized sources, a phenomenon relevant to thermal analysis, anisotropic materials, and environmental modeling.https://bst.jurnal.unej.ac.id/index.php/BST/article/view/53711anisotropic diffusionfinite element methodimplicit and explicit euler schemes |
| spellingShingle | M. Ziaul Arif Millatuz Zahroh Sailah Ar Rizka Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method Berkala Sainstek anisotropic diffusion finite element method implicit and explicit euler schemes |
| title | Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method |
| title_full | Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method |
| title_fullStr | Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method |
| title_full_unstemmed | Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method |
| title_short | Numerical Solution of Anisotropic Diffusion with Localized Source Using Euler Scheme and Finite Element Method |
| title_sort | numerical solution of anisotropic diffusion with localized source using euler scheme and finite element method |
| topic | anisotropic diffusion finite element method implicit and explicit euler schemes |
| url | https://bst.jurnal.unej.ac.id/index.php/BST/article/view/53711 |
| work_keys_str_mv | AT mziaularif numericalsolutionofanisotropicdiffusionwithlocalizedsourceusingeulerschemeandfiniteelementmethod AT millatuzzahroh numericalsolutionofanisotropicdiffusionwithlocalizedsourceusingeulerschemeandfiniteelementmethod AT sailaharrizka numericalsolutionofanisotropicdiffusionwithlocalizedsourceusingeulerschemeandfiniteelementmethod |