  We develop and analyze a class of maximum bound preserving schemes for
approximately solving Allen--Cahn equations. We apply a $k$th-order single-step
scheme in time (where the nonlinear term is linearized by multi-step
extrapolation), and a lumped mass finite element method in space with piecewise
$r$th-order polynomials and Gauss--Lobatto quadrature. At each time level, a
cut-off post-processing is proposed to eliminate extra values violating the
maximum bound principle at the finite element nodal points. As a result, the
numerical solution satisfies the maximum bound principle (at all nodal points),
and the optimal error bound $O(\tau^k+h^{r+1})$ is theoretically proved for a
certain class of schemes. These time stepping schemes under consideration
includes algebraically stable collocation-type methods, which could be
arbitrarily high-order in both space and time. Moreover, combining the cut-off
strategy with the scalar auxiliary value (SAV) technique, we develop a class of
energy-stable and maximum bound preserving schemes, which is arbitrarily
high-order in time. Numerical results are provided to illustrate the accuracy
of the proposed method.
