Unconditionally Maximum-Principle-Preserving Parametric Integrating Factor Two-Step Runge-Kutta Schemes for Parabolic Sine-Gordon Equations (CSIAM Transactions on Applied Mathematics, 2023)
We present a systematic two-step approach to derive temporal up to theeighth-order, unconditionally maximum-principle-preserving schemes for a semilinear parabolic sine-Gordon equation and its conservative modification. By introducinga stabilization term to an explicit integrating factor approach, and designing suitableapproximations 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 theButcher coefficients and non-decreasing constraint of the abscissas. The order conditions, linear stability, and convergence in the $L^∞$-norm are analyzed. Theoretical andnumerical results demonstrate that the proposed framework, which is explicit and freeof limiters, cut-off post-processing, or exponential effects, offers a concise, and effectiveapproach to develop high-order inequality-preserving and linear-invariant-conservingalgorithms.
Keywords: Parabolic sine-Gordon equation; linear-invariant-conserving; unconditionally maximum-principle-preserving; parametric two-step Runge-Kutta method.