Hybrid Explicit-Implicit Predictor-Corrector Exponential Time-Differencing Multistep Pad\'{e} Schemes for Semilinear Parabolic Equations with Time-Delay

In this paper, we propose and analyze ETD-Multistep-Pad\'{e} (ETD-MS-Pad\'{e}) and ETD Implicit Multistep-Pad\'{e} (ETD-IMS-Pad\'{e}) for semilinear parabolic delay differential equations with smooth solutions. In our previous work [15], we proposed ETD-RK-Pad\'{e} 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-Pad\'{e} and ETD-IMS-Pad\'{e} schemes for the solution of semilinear parabolic equations with delay. We synergize the ETD-MS-Pad\'{e} with ETD-IMS-Pad\'{e} 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 Pad\'{e} approach to approximate this exponential operator. This methodology leads to the development of the ETD-MS-Pad\'{e} and ETD-IMS-Pad\'{e} schemes, applicable even for arbitrary time orders. We validate the ETD-MS1,2,3,4-Pad\'{e} schemes and ETD-IMS2,3,4 schemes through numerical experiments.