  In general, high order splitting methods suffer from an order reduction
phenomena when applied to the time integration of partial differential
equations with non-periodic boundary conditions. In the last decade, there were
introduced several modifications to prevent the second order Strang Splitting
method from such a phenomena. In this article, inspired by these recent
corrector techniques, we introduce a splitting method of order three for a
class of semilinear parabolic problems that avoids order reduction in the
context of non-periodic boundary conditions. We give a proof for the third
order convergence of the method in a simplified linear setting and confirm the
result by numerical experiments. Moreover, we show numerically that the high
order convergence persists for an order four variant of a splitting method, and
also for a nonlinear source term.
