The numerical solutions of linear semi-discrete evolution problems on the half-line using the Unified Transform Method

TL;DR

This paper extends the Unified Transform Method to semi-discrete linear evolution problems on the half-line, providing exact solutions, analyzing boundary conditions, and demonstrating numerical applications.

Contribution

It introduces a semi-discrete analogue of the Unified Transform Method for linear PDEs, enabling exact solutions for discretized problems on the half-line.

Findings

Exact semi-discrete solutions for linear evolution equations

Analysis of boundary conditions and ghost points

Numerical examples demonstrating method effectiveness

Abstract

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.