Exp-ParaDiag: Time-Parallel Exponential Integrators for Parabolic PDEs

TL;DR

Exp-ParaDiag introduces a novel time-parallel exponential integrator framework for parabolic PDEs, achieving high-order accuracy and demonstrating robustness and efficiency through rigorous analysis and numerical experiments.

Contribution

It develops and analyzes a new time-parallel exponential integrator method, Exp-ParaDiag, with high-order accuracy and applicability to nonlinear problems.

Findings

Convergence established for fixed-point and GMRES iterations.

Achieves sixth-order temporal accuracy.

Numerical experiments confirm robustness and efficiency.

Abstract

This paper introduces Exp-ParaDiag, a novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework. We develop and analyze Exp-ParaDiag based on first and second order accurate exponential integrators. We establish the convergence of the proposed methods both as preconditioned fixed-point iterations and as precon- ditioners within the GMRES framework. Furthermore, we extend the Exp-ParaDiag formulation to achieve sixth-order temporal accuracy using exponential integrators. The proposed approach is also generalized to nonlinear problems, for which convergence is rigorously demonstrated. A series of numerical experiments is presented to validate the theoretical results and to illustrate the robustness and efficiency of the developed methods.