Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913627507785728 |
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| author | Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei |
| author_facet | Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei |
| contents | Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the potential formation of space-filling quasiperiodic structures that lack translational symmetry or decay. In this paper, we introduce a highly accurate numerical method for solving time-dependent quasiperiodic parabolic equations. We discretize the spatial variables using the projection method (PM) and the time variable with the second-order backward differentiation formula (BDF2). We provide a complexity analysis for the resulting PM-BDF2 method. Furthermore, we conduct a detailed convergence analysis, demonstrating that the proposed method exhibits spectral accuracy in space and second-order accuracy in time. Numerical results in both one and two dimensions validate these convergence results, highlighting the PM-BDF2 method as a highly efficient algorithm for addressing quasiperiodic parabolic equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19175 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei Numerical Analysis Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the potential formation of space-filling quasiperiodic structures that lack translational symmetry or decay. In this paper, we introduce a highly accurate numerical method for solving time-dependent quasiperiodic parabolic equations. We discretize the spatial variables using the projection method (PM) and the time variable with the second-order backward differentiation formula (BDF2). We provide a complexity analysis for the resulting PM-BDF2 method. Furthermore, we conduct a detailed convergence analysis, demonstrating that the proposed method exhibits spectral accuracy in space and second-order accuracy in time. Numerical results in both one and two dimensions validate these convergence results, highlighting the PM-BDF2 method as a highly efficient algorithm for addressing quasiperiodic parabolic equations. |
| title | Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2412.19175 |