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

TL;DR

This paper introduces new explicit-implicit predictor-corrector exponential time-differencing multistep schemes with Padé approximation for solving semilinear parabolic delay differential equations, demonstrating improved convergence and efficiency.

Contribution

The paper develops and analyzes ETD-MS-Padé and ETD-IMS-Padé schemes, providing a simpler, efficient predictor-corrector approach for semilinear parabolic equations with delay, extending previous complex methods.

Findings

Better convergence compared to EERK scheme

Effective for arbitrary time orders

Validated through numerical experiments

Abstract

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.