Hybrid Explicit-Implicit Predictor-Corrector Exponential Time-Differencing Multistep Padé Schemes for Semilinear Parabolic Equations with Time-Delay
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| Format: | Preprint |
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2025
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| _version_ | 1866908429254131712 |
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| author | Dai, Haishen Lei, Huan |
| author_facet | Dai, Haishen Lei, Huan |
| contents | In this paper, we propose and analyze ETD-Multistep-Padé (ETD-MS-Padé) and ETD Implicit Multistep-Padé (ETD-IMS-Padé) for semilinear parabolic delay differential equations with smooth solutions. In our previous work [15], we proposed ETD-RK-Padé scheme to compute high-order numerical solutions for nonlinear parabolic reaction-diffusion equation with constant time delay. However, the based ETD-RK numerical scheme in [15] is very complex and the corresponding calculation program is also very complicated. We propose in this paper ETD-MS-Padé and ETD-IMS-Padé schemes for the solution of semilinear parabolic equations with delay. We synergize the ETD-MS-Padé with ETD-IMS-Padé to construct efficient predictor-corrector scheme. This new predictor-corrector scheme will become an important tool for solving the numerical solutions of parabolic differential equations. Remarkably, we also conducted experiments in Table$10$ to compare the numerical results of the predictor-corrector scheme with the EERK scheme proposed in paper [42]. The predictor-corrector scheme demonstrated better convergence.
The main idea is to employ an ETD-based Adams multistep extrapolation for the time integration of the corresponding equation. To overcome the well-known numerical instability associated with computing the exponential operator, we utilize the Padé approach to approximate this exponential operator. This methodology leads to the development of the ETD-MS-Padé and ETD-IMS-Padé schemes, applicable even for arbitrary time orders. We validate the ETD-MS1,2,3,4-Padé schemes and ETD-IMS2,3,4 schemes through numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22664 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hybrid Explicit-Implicit Predictor-Corrector Exponential Time-Differencing Multistep Padé Schemes for Semilinear Parabolic Equations with Time-Delay Dai, Haishen Lei, Huan Numerical Analysis In this paper, we propose and analyze ETD-Multistep-Padé (ETD-MS-Padé) and ETD Implicit Multistep-Padé (ETD-IMS-Padé) for semilinear parabolic delay differential equations with smooth solutions. In our previous work [15], we proposed ETD-RK-Padé scheme to compute high-order numerical solutions for nonlinear parabolic reaction-diffusion equation with constant time delay. However, the based ETD-RK numerical scheme in [15] is very complex and the corresponding calculation program is also very complicated. We propose in this paper ETD-MS-Padé and ETD-IMS-Padé schemes for the solution of semilinear parabolic equations with delay. We synergize the ETD-MS-Padé with ETD-IMS-Padé to construct efficient predictor-corrector scheme. This new predictor-corrector scheme will become an important tool for solving the numerical solutions of parabolic differential equations. Remarkably, we also conducted experiments in Table$10$ to compare the numerical results of the predictor-corrector scheme with the EERK scheme proposed in paper [42]. The predictor-corrector scheme demonstrated better convergence. The main idea is to employ an ETD-based Adams multistep extrapolation for the time integration of the corresponding equation. To overcome the well-known numerical instability associated with computing the exponential operator, we utilize the Padé approach to approximate this exponential operator. This methodology leads to the development of the ETD-MS-Padé and ETD-IMS-Padé schemes, applicable even for arbitrary time orders. We validate the ETD-MS1,2,3,4-Padé schemes and ETD-IMS2,3,4 schemes through numerical experiments. |
| title | Hybrid Explicit-Implicit Predictor-Corrector Exponential Time-Differencing Multistep Padé Schemes for Semilinear Parabolic Equations with Time-Delay |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.22664 |