The numerical solution of semidiscrete linear evolution problems on the finite interval using the Unified Transform Method

TL;DR

This paper extends the Unified Transform Method to semidiscrete linear evolution PDEs on finite intervals, providing exact solutions, numerical representations, and insights into boundary conditions and discretizations.

Contribution

It introduces a semidiscrete version of the Unified Transform Method for linear PDEs, analyzing boundary conditions, discretizations, and continuum limits.

Findings

Exact semidiscrete solutions for linear evolution equations.

Alternative series for improved numerical computation.

Analysis of boundary conditions and discretization effects.

Abstract

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.