Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains
Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tra...
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MDPI AG
2023-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/11/823 |
| _version_ | 1797459281103028224 |
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| author | Zhipeng Li Hongwu Tang Saiyu Yuan Huiming Zhang Lingzhong Kong HongGuang Sun |
| author_facet | Zhipeng Li Hongwu Tang Saiyu Yuan Huiming Zhang Lingzhong Kong HongGuang Sun |
| author_sort | Zhipeng Li |
| collection | DOAJ |
| description | Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long distance in either the forward or backward direction along preferential channels during the transport. The classical advection–diffusion (ADE) model has been widely used to describe tracer transport; however, they often fall short in capturing the intricacies of nonlocal diffusion processes. The fractional operator has gained recognition among hydrologists due to its potential to capture distinct mechanisms of transport and storage for tracer particles exhibiting nonlocal dynamics. However, the hypersingularity of the fractional Laplacian operator presents considerable difficulties in its numerical approximation in bounded domains. This study focuses on the development and application of the fractional Laplacian-based model to characterize nonlocal tracer transport behavior, specifically considering both forward and backward diffusion processes on bounded domains. The Riesz fractional Laplacian provides a mathematical framework for describing tracer diffusion processes on a bounded domain, and a novel finite difference method (FDM) is introduced as an effective numerical solver for addressing the fractional Laplacian-based model. Applications reveal that the fractional Laplacian-based model can effectively capture the observed nonlocal tracer transport behavior in a heterogeneous system, and nonlocal tracer transport exhibits dynamic characteristics, influenced by the evolving heterogeneity of the media at various temporal scales. |
| first_indexed | 2024-03-09T16:49:12Z |
| format | Article |
| id | doaj.art-b2b5785197504c1ea4b675749ab42b86 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T16:49:12Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-b2b5785197504c1ea4b675749ab42b862023-11-24T14:43:06ZengMDPI AGFractal and Fractional2504-31102023-11-0171182310.3390/fractalfract7110823Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded DomainsZhipeng Li0Hongwu Tang1Saiyu Yuan2Huiming Zhang3Lingzhong Kong4HongGuang Sun5National Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, ChinaNational Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, ChinaNational Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, ChinaNational Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, ChinaNational Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, ChinaNational Key Laboratory of Water Disaster Prevention, Hohai University, Nanjing 210098, ChinaRecent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long distance in either the forward or backward direction along preferential channels during the transport. The classical advection–diffusion (ADE) model has been widely used to describe tracer transport; however, they often fall short in capturing the intricacies of nonlocal diffusion processes. The fractional operator has gained recognition among hydrologists due to its potential to capture distinct mechanisms of transport and storage for tracer particles exhibiting nonlocal dynamics. However, the hypersingularity of the fractional Laplacian operator presents considerable difficulties in its numerical approximation in bounded domains. This study focuses on the development and application of the fractional Laplacian-based model to characterize nonlocal tracer transport behavior, specifically considering both forward and backward diffusion processes on bounded domains. The Riesz fractional Laplacian provides a mathematical framework for describing tracer diffusion processes on a bounded domain, and a novel finite difference method (FDM) is introduced as an effective numerical solver for addressing the fractional Laplacian-based model. Applications reveal that the fractional Laplacian-based model can effectively capture the observed nonlocal tracer transport behavior in a heterogeneous system, and nonlocal tracer transport exhibits dynamic characteristics, influenced by the evolving heterogeneity of the media at various temporal scales.https://www.mdpi.com/2504-3110/7/11/823fractional Laplacianforward and backward diffusionnonlocal modelheterogeneous systemsfinite difference method |
| spellingShingle | Zhipeng Li Hongwu Tang Saiyu Yuan Huiming Zhang Lingzhong Kong HongGuang Sun Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains Fractal and Fractional fractional Laplacian forward and backward diffusion nonlocal model heterogeneous systems finite difference method |
| title | Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains |
| title_full | Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains |
| title_fullStr | Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains |
| title_full_unstemmed | Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains |
| title_short | Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains |
| title_sort | modeling long distance forward and backward diffusion processes in tracer transport using the fractional laplacian on bounded domains |
| topic | fractional Laplacian forward and backward diffusion nonlocal model heterogeneous systems finite difference method |
| url | https://www.mdpi.com/2504-3110/7/11/823 |
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