Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation
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| Format: | Preprint |
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2025
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| _version_ | 1866909859271671808 |
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| author | Zhang, Bingyin Fu, Hongfei Lan, Rihui Xie, Shusen |
| author_facet | Zhang, Bingyin Fu, Hongfei Lan, Rihui Xie, Shusen |
| contents | In this paper, we propose and analyze a linear, structure-preserving scalar auxiliary variable (SAV) method for solving the Allen--Cahn equation based on the second-order backward differentiation formula (BDF2) with variable time steps. To this end, we first design a novel and essential auxiliary functional that serves twofold functions: (i) ensuring that a first-order approximation to the auxiliary variable, which is essentially important for deriving the unconditional energy dissipation law, does not affect the second-order temporal accuracy of the phase function $ϕ$; and (ii) allowing us to develop effective stabilization terms that are helpful to establish the MBP-preserving linear methods. Together with this novel functional and standard central difference stencil, we then propose a linear, second-order variable-step BDF2 type stabilized exponential SAV scheme, namely BDF2-sESAV-I, which is shown to preserve both the discrete modified energy dissipation law under the temporal stepsize ratio $ 0 < r_{k} := τ_{k}/τ_{k-1} < 4.864 - δ$ with a positive constant $δ$ and the MBP under $ 0 < r_{k} < 1 + \sqrt{2} $. Moreover, an analysis of the approximation to the original energy by the modified one is presented. With the help of the kernel recombination technique, optimal $ H^{1}$- and $ L^{\infty}$-norm error estimates of the variable-step BDF2-sESAV-I scheme are rigorously established. Numerical examples are carried out to verify the theoretical results and demonstrate the effectiveness and efficiency of the proposed scheme. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_04616 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation Zhang, Bingyin Fu, Hongfei Lan, Rihui Xie, Shusen Numerical Analysis In this paper, we propose and analyze a linear, structure-preserving scalar auxiliary variable (SAV) method for solving the Allen--Cahn equation based on the second-order backward differentiation formula (BDF2) with variable time steps. To this end, we first design a novel and essential auxiliary functional that serves twofold functions: (i) ensuring that a first-order approximation to the auxiliary variable, which is essentially important for deriving the unconditional energy dissipation law, does not affect the second-order temporal accuracy of the phase function $ϕ$; and (ii) allowing us to develop effective stabilization terms that are helpful to establish the MBP-preserving linear methods. Together with this novel functional and standard central difference stencil, we then propose a linear, second-order variable-step BDF2 type stabilized exponential SAV scheme, namely BDF2-sESAV-I, which is shown to preserve both the discrete modified energy dissipation law under the temporal stepsize ratio $ 0 < r_{k} := τ_{k}/τ_{k-1} < 4.864 - δ$ with a positive constant $δ$ and the MBP under $ 0 < r_{k} < 1 + \sqrt{2} $. Moreover, an analysis of the approximation to the original energy by the modified one is presented. With the help of the kernel recombination technique, optimal $ H^{1}$- and $ L^{\infty}$-norm error estimates of the variable-step BDF2-sESAV-I scheme are rigorously established. Numerical examples are carried out to verify the theoretical results and demonstrate the effectiveness and efficiency of the proposed scheme. |
| title | Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2502.04616 |