The numerical solutions of linear semidiscrete evolution problems on the half‐line using the Unified Transform Method

Abstract

AbstractWe discuss a semidiscrete analogue of the Unified Transform Method (UTM), introduced by A. S. Fokas, to solve initial‐boundary‐value problems for linear evolution partial differential equations of constant coefficients. The semidiscrete method is applied to various spacial discretizations of several first‐ and second‐order linear equations on the half‐line , producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. We additionally show how the UTM treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and propose the notion of “natural” discretizations. We consider the continuum limit of the semidiscrete solutions and provide several numerical examples.