  We study a semidiscrete analogue of the Unified Transform Method introduced
by A. S. Fokas, to solve initial-boundary-value problems for linear evolution
partial differential equations with constant coefficients on the finite
interval $x \in (0,L)$. The semidiscrete method is applied to various spatial
discretizations of several first and second-order linear equations, producing
the exact solution for the semidiscrete problem, given appropriate initial and
boundary data. From these solutions, we derive alternative series
representations that are better suited for numerical computations. In addition,
we show how the Unified Transform Method treats derivative boundary conditions
and ghost points introduced by the choice of discretization stencil and we
propose the notion of "natural" discretizations. We consider the continuum
limit of the semidiscrete solutions and compare with standard finite-difference
schemes.
