  The energy dissipation law and the maximum bound principle are two critical
physical properties of the Allen--Cahn equations. While many existing
time-stepping methods are known to preserve the energy dissipation law, most
apply to a modified form of energy. In this work, we demonstrate that, when the
nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of
arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK)
schemes preserve the original energy dissipation property, under a mild
step-size constraint. Additionally, we guarantee the Lipschitz condition on the
nonlinear term by applying a rescaling post-processing technique, which ensures
that the numerical solution unconditionally satisfies the maximum bound
principle. Consequently, our proposed schemes maintain both the original energy
dissipation law and the maximum bound principle and can achieve arbitrarily
high-order accuracy. We also establish an optimal error estimate for the
proposed schemes. Some numerical experiments are carried out to verify our
theoretical results.
