  We discuss a semi-discrete analogue of the Unified Transform Method,
introduced by A. S. Fokas, to solve initial-boundary-value problems for linear
evolution partial differential equations of constant coefficients. The
semi-discrete method is applied to various spacial discretizations of several
first and second-order linear equations on the half-line $x \geq 0$, producing
the exact solution for the semi-discrete problem, given appropriate initial and
boundary data. We additionally show how the Unified Transform Method treats
derivative boundary conditions and ghost points introduced by the choice of
discretization stencil. We consider the continuum limit of the semi-discrete
solutions and provide several numerical examples.
