A second-order dynamical low-rank mass-lumped finite element method for the Allen-Cahn equation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915097489702912 |
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| author | Yang, Jun Yi, Nianyu Yin, Peimeng |
| author_facet | Yang, Jun Yi, Nianyu Yin, Peimeng |
| contents | In this paper, we propose a novel second-order dynamical low-rank mass-lumped finite element method for solving the Allen-Cahn (AC) equation, a semilinear parabolic partial differential equation. The matrix differential equation of the semi-discrete mass-lumped finite element scheme is decomposed into linear and nonlinear components using the second-order Strang splitting method. The linear component is solved analytically within a low-rank manifold, while the nonlinear component is discretized using a second-order augmented basis update & Galerkin (BUG) integrator, in which the $S$-step matrix equation is solved by the explicit 2-stage strong stability-preserving Runge-Kutta method. The algorithm has lower computational complexity than the full-rank mass-lump finite element method. The dynamical low-rank finite element solution is shown to conserve mass up to a truncation tolerance for the conservative Allen-Cahn equation. Meanwhile, the modified energy is dissipative up to a high-order error and is hence stable. Numerical experiments validate the theoretical results. Symmetry-preserving tests highlight the robustness of the proposed method for long-time simulations and demonstrate its superior performance compared to existing methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A second-order dynamical low-rank mass-lumped finite element method for the Allen-Cahn equation Yang, Jun Yi, Nianyu Yin, Peimeng Numerical Analysis 35K58, 65F55, 65M60, 65Y20 In this paper, we propose a novel second-order dynamical low-rank mass-lumped finite element method for solving the Allen-Cahn (AC) equation, a semilinear parabolic partial differential equation. The matrix differential equation of the semi-discrete mass-lumped finite element scheme is decomposed into linear and nonlinear components using the second-order Strang splitting method. The linear component is solved analytically within a low-rank manifold, while the nonlinear component is discretized using a second-order augmented basis update & Galerkin (BUG) integrator, in which the $S$-step matrix equation is solved by the explicit 2-stage strong stability-preserving Runge-Kutta method. The algorithm has lower computational complexity than the full-rank mass-lump finite element method. The dynamical low-rank finite element solution is shown to conserve mass up to a truncation tolerance for the conservative Allen-Cahn equation. Meanwhile, the modified energy is dissipative up to a high-order error and is hence stable. Numerical experiments validate the theoretical results. Symmetry-preserving tests highlight the robustness of the proposed method for long-time simulations and demonstrate its superior performance compared to existing methods. |
| title | A second-order dynamical low-rank mass-lumped finite element method for the Allen-Cahn equation |
| topic | Numerical Analysis 35K58, 65F55, 65M60, 65Y20 |
| url | https://arxiv.org/abs/2501.06145 |