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.

翻译：我们讨论由A.S.Fokas介绍的统一变换方法的半分解类比,以解决线性进化中不变系数部分差异方程式的初始界限值问题。半分解方法适用于半线美元=Geq 0美元上若干一级和二级一级线性方程式的片段分解,根据适当的初始数据和边界数据,为半分解问题提供确切的解决方案。我们还演示了统一变换方法如何处理衍生的边界条件和因选择分解加速器而引入的幽灵点。我们考虑了半分解办法的连续限制,并提供了几个数字例子。