Unconditionally Maximum-Principle-Preserving Parametric Integrating Factor Two-Step Runge-Kutta Schemes for Parabolic Sine-Gordon Equations

Abstract

We present a systematic two-step approach to derive temporal up to the eighth-order, unconditionally maximum-principle-preserving schemes for a semilinear parabolic sine-Gordon equation and its conservative modification. By introducing a stabilization term to an explicit integrating factor approach, and designing suitable approximations to the exponential functions, we propose a unified parametric two-step Runge-Kutta framework to conserve the linear invariant of the original system. To preserve the maximum principle unconditionally, we develop parametric integrating factor two-step Runge-Kutta schemes by enforcing the non-negativeness of the Butcher coefficients and non-decreasing constraint of the abscissas. The order conditions, linear stability, and convergence in the $L^∞$-norm are analyzed. Theoretical and numerical results demonstrate that the proposed framework, which is explicit and free of limiters, cut-off post-processing, or exponential effects, offers a concise, and effective approach to develop high-order inequality-preserving and linear-invariant-conserving algorithms.