  This paper considers the numerical integration of semilinear evolution PDEs
using the high order linearly implicit methods developped in a previous paper
in the ODE setting. These methods use a collocation Runge--Kutta method as a
basis, and additional variables that are updated explicitly and make the
implicit part of the collocation Runge--Kutta method only linearly implicit. In
this paper, we introduce several notions of stability for the underlying
Runge--Kutta methods as well as for the explicit step on the additional
variables necessary to fit the context of evolution PDE. We prove a main
theorem about the high order of convergence of these linearly implicit methods
in this PDE setting, using the stability hypotheses introduced before. We use
nonlinear Schr\''odinger equations and heat equations as main examples but our
results extend beyond these two classes of evolution PDEs. We illustrate our
main result numerically in dimensions 1 and 2, and we compare the efficiency of
the linearly implicit methods with other methods from the litterature. We also
illustrate numerically the necessity of the stability conditions of our main
result.
