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 α i , j , nondecreasing abscissas c i and the time step size τ > 0 satisfies τ { β i , j α i , j } ∈ [ − 4 , 1 2 ] . We prove that the proposed method is convergent with order O ( τ p + h 2 ) 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.