  We propose two efficient energetic spectral-element methods in time for
marching nonlinear gradient systems with the phase-field Allen--Cahn equation
as an example: one fully implicit nonlinear method and one semi-implicit linear
method. Different from other spectral methods in time using spectral
Petrov-Galerkin or weighted Galerkin approximations, the presented implicit
method employs an energetic variational Galerkin form that can maintain the
mass conservation and energy dissipation property of the continuous dynamical
system. Another advantage of this method is its superconvergence. A high-order
extrapolation is adopted for the nonlinear term to get the semi-implicit
method. The semi-implicit method does not have superconvergence, but can be
improved by a few Picard-like iterations to recover the superconvergence of the
implicit method. Numerical experiments verify that the method using Legendre
elements of degree three outperforms the 4th-order implicit-explicit backward
differentiation formula and the 4th-order exponential time difference
Runge-Kutta method, which were known to have best performances in solving
phase-field equations. In addition to the standard Allen--Cahn equation, we
also apply the method to a conservative Allen--Cahn equation, in which the
conservation of discrete total mass is verified. The applications of the
proposed methods are not limited to phase-field Allen--Cahn equations. They are
suitable for solving general, large-scale nonlinear dynamical systems.
