Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation

Abstract

Whether high order temporal integrators can preserve the maximum principle of Allen-Cahn equation has been an open problem in recent years. This work provides a positive answer by designing and analyzing a class of up to fourth order maximum principle preserving integrators for the Allen-Cahn equation. First, the second order finite difference discretization is applied to the Allen-Cahn equation in the space direction. The obtained semi-discrete system also preserves the maximum principle and the energy dissipation law. Then the fully discrete numerical scheme is obtained by applying the Lawson transformation and the Runge-Kutta integration in the time direction. We define sufficient conditions for explicit integration factor Runge-Kutta scheme to preserve the maximum principle, namely, the Shu-Osher form of the underlying Runge-Kutta scheme has non-negative coefficients αi,j, nondecreasing abscissas ci and the time step size τ>0 satisfies τ{βi,jαi,j}∈[−4,12]. We prove that the proposed method is convergent with order O(τp+h2) in the discrete L∞ norm. A fast solver is then applied to the discrete system to accelerate numerical computations. Various experiments for 1D, 2D and 3D problems are provided to illustrate the high-order convergence and maximum principle preserving of the proposed algorithms over a long time and verify the theoretical analysis.