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

2021/03/22 by Jorge Cisneros, Cisneros, Jorge, Bernard Deconinck +1
Mathematics · #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2103.12163

openalex publication_date 2021/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We discuss a semi-discrete analogue of the Unified Transform Method,\nintroduced by A. S. Fokas, to solve initial-boundary-value problems for linear\nevolution partial differential equations of constant coefficients. The\nsemi-discrete method is applied to various spacial discretizations of several\nfirst and second-order linear equations on the half-line x \≥ 0, producing\nthe exact solution for the semi-discrete problem, given appropriate initial and\nboundary data. We additionally show how the Unified Transform Method treats\nderivative boundary conditions and ghost points introduced by the choice of\ndiscretization stencil. We consider the continuum limit of the semi-discrete\nsolutions and provide several numerical examples.\n

Related

• The numerical solution of semidiscrete linear evolution problems on the finite interval using the Unified Transform Method
• The Numerical Unified Transform Method for Initial-boundary Value Problems on the Half-line
• The unified transform method for linear initial-boundary value problems: a spectral interpretation
• Generalized Dirichlet to Neumann Maps for Linear Dispersive Equations on Half-Line
• Infinity of solutions to initial-boundary value problems for linear constant-coefficient evolution PDEs on semi-infinite intervals