  Maximum bound principle (MBP) is an important property for a large class of
semilinear parabolic equations, in the sense that the time-dependent solution
of the equation with appropriate initial and boundary conditions and nonlinear
operator preserves for all time a uniform pointwise bound in absolute value. It
has been a challenging problem on how to design unconditionally MBP-preserving
high-order accurate time-stepping schemes for these equations. In this paper,
we combine the integrating factor Runge-Kutta (IFRK) method with the linear
stabilization technique to develop a stabilized IFRK (sIFRK) method, and
successfully derive sufficient conditions for the proposed method to preserve
MBP unconditionally in the discrete setting. We then elaborate some sIFRK
schemes with up to the third-order accuracy, which are proven to be
unconditionally MBP-preserving by verifying these conditions. In addition, it
is shown that many classic strong stability-preserving sIFRK schemes do not
satisfy these conditions except the first-order one. Extensive numerical
experiments are also carried out to demonstrate the performance of the proposed
method.
