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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.11175 |
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| _version_ | 1866916106157948928 |
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| author | Jacobson, Alon Hu, Xiaozhe |
| author_facet | Jacobson, Alon Hu, Xiaozhe |
| contents | Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties $\operatorname{curl}^α\operatorname{grad}^α= \mathbf{0}$ and $\operatorname{div}^α\operatorname{curl}^α= 0$ hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11175 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus Jacobson, Alon Hu, Xiaozhe Numerical Analysis 65M99 65N99 26A33 35R11 Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties $\operatorname{curl}^α\operatorname{grad}^α= \mathbf{0}$ and $\operatorname{div}^α\operatorname{curl}^α= 0$ hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size. |
| title | Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus |
| topic | Numerical Analysis 65M99 65N99 26A33 35R11 |
| url | https://arxiv.org/abs/2210.11175 |