Maximum bound principle and original energy dissipation of arbitrarily high-order ETD Runge–Kutta schemes for Allen–Cahn equations

The energy dissipation law and the maximum bound principle are two fundamental physical properties of the Allen–Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law most of them apply to a modified form of energy. In this work we show that, when the nonlinear term of the Allen–Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge–Kutta schemes with the linear stabilization technique preserves the original energy dissipation property under some step-size constraint. To ensure the Lipschitz condition on the nonlinear term we introduce a rescaling post-processing technique, which guarantees that the numerical solution unconditionally satisfies the maximum bound principle. As a result, our proposed schemes simultaneously maintain both the original energy dissipation law and the maximum bound principle while achieving arbitrarily high-order accuracy. We also establish the optimal error estimate for the proposed schemes. Numerical experiments fully confirm the convergence rates, the preservation of the maximum bound principle and the original energy dissipation property, as well as the high efficiency of the high-order schemes for long-time simulations.