The Unified Transform Method and its semi-discrete analogue for numerically solving IBVPs

Abstract

Finite-difference schemes are a popular and intuitive approach to numerically solve nonlinear initial-boundary value problems (IBVPs). Often, this leads to the introduction of ghost points, where the numerical method depends on grid points outside of the working domain. The usual heuristics of doing this for second-order problems do not generalize to higher order, and incorporating boundary conditions and addressing ghost points are serious numerical issues. Our approach proposes to tackle this problem by the implementation of split-step methods to separately solve the linear and nonlinear subproblems. In this dissertation, we discuss the Unified Transform Method (UTM), introduced by A. S. Fokas, and its semi-discrete analogue to devise finite-difference schemes for the linear problem that appropriately incorporate boundary conditions. The UTM solution representations are then treated to give analytic continuation formulas that can be applied at ghost points in the split-step method. We present our developments through examples of several linear problems and their discretizations on the half-line and finite interval, and the nonlinear Schrödinger equation on the finite interval. We discuss the continuum limit of the solutions and numerical results.