Implicit-explicit time discretization with complex time for parabolic equations

This paper presents a novel high-order time discretization scheme on the time complex plane for parabolic equations. The innovation of this method lies in its ability to straightforwardly construct temporal discretization schemes of arbitrary order while achieving stability comparable to explicit Runge-Kutta methods, and simultaneously enhancing computational efficiency without compromising stability. By employing a matrix construction method, we unify the characterization of the stability region for both ordinary differential equations and partial differential equations. We prove the L ² stability and arbitrary-order convergence of the method. Numerical experiments demonstrate that, compared to explicit Runge-Kutta methods, the proposed scheme improves computational efficiency. This method offers an efficient and accurate solution for applications with stringent precision.