Quantifying and eliminating the time delay in stabilization exponential time differencing Runge--Kutta schemes for the Allen--Cahn equation

Abstract

Although the stabilization technique is favorable in designing unconditionally stable schemes for stiff and nonlinear systems, the induced time delay is intractable in computations. In this paper, we propose a class of delay-free stabilization schemes for the Allen--Cahn gradient flow system. Considering the Fourier pseudo-spectral spatial discretization for the Allen--Cahn equation with either the polynomial or the logarithmic potential, we establish a semi-discrete, mesh-dependent maximum principle by adopting a stabilization technique. To unconditionally preserve the maximum principle and energy stability, we investigate a family of exponential time differencing Runge--Kutta (ETDRK) integrators up to the second-order. After reformulating the ETDRK schemes as a class of parametric Runge--Kutta integrators, we quantify the lagging effect brought by stabilization, and eliminate delayed convergence using a relaxation technique. The temporal error estimate of the relaxation ETDRK integrators in the maximum norm topology is analyzed under a fixed spatial mesh. Numerical experiments demonstrate the delay-free and structure-preserving properties of the proposed schemes.