  This paper introduces a new class of numerical methods for the time
integration of evolution equations set as Cauchy problems of ODEs or PDEs. The
systematic design of these methods mixes the Runge-Kutta collocation formalism
with collocation techniques, in such a way that the methods are linearly
implicit and have high order. The fact that these methods are implicit allows
to avoid CFL conditions when the large systems to integrate come from the space
discretization of evolution PDEs. Moreover, these methods are expected to be
efficient since they only require to solve one linear system of equations at
each time step, and efficient techniques from the literature can be used to do
so. After the introduction of the methods, we set suitable definitions of
consistency and stability for these methods. This allows for a proof that
arbitrarily high order linearly implicit methods exist and converge when
applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs
that illustrate our theoretical results for ODEs, and compare our methods with
standard methods for several evolution PDEs.
