Third-order accurate, large time-stepping and maximum-principle-preserving schemes for the Allen-Cahn equation

We present and evaluate several explicit, large time-stepping algorithms for the Allen Cahn equation. Our approach incorporates a stabilization technique and uses Taylor series approximations for exponential functions to develop a family of up to third-order parametric Runge-Kutta schemes that maintain fixed-points and maximum principle for any time step t > 0. We also introduce a new relaxation technique that eliminates time delay caused by stabilization. To further decrease the stabilization parameter, we utilize an integrating factor with respect to the stiff linear operator and develop a parametric relaxation integrating factor Runge-Kutta (pRIFRK) framework. Compared to existing maximum-principle-preserving (MPP) schemes, the proposed parametric relaxation approaches are free from limiters, cut-off post-processing, exponential decay, or time delay. Linear stability analysis determines that the parametric approaches are A-stable when appropriate parameters are used. In addition, we provide error estimates in the l(8)-norm with the help of the MPP property. We demonstrate the high-order temporal accuracy, maximum-principle-preservation, energy stability, and delay-free properties of the proposed schemes through a set of experiments on 1D, 2D, and 3D problems.