The Numerical Unified Transform Method for Initial-boundary Value Problems on the Half-line

TL;DR

This paper introduces a numerical implementation of the Unified Transform Method for solving linear PDEs on the half-line, achieving high accuracy without spatial discretization or time stepping, even for large x and t.

Contribution

The paper presents a novel numerical approach based on the Fokas method that avoids traditional discretization, enabling efficient and accurate solutions over large domains.

Findings

Method maintains high accuracy for large x and t

No assumptions on initial or boundary functions needed

Complexity remains stable regardless of x,t size

Abstract

We implement the Unified Transform Method of Fokas as a numerical method to solve linear partial differential equations on the half-line. The method computes the solution at any x and t without spatial discretization or time stepping. With the help of contour deformations and oscillatory integration techniques, the method's complexity does not increase for large x,t and the method is more accurate as x,t increase. Our goal is to make no assumptions on the functional form of the initial or boundary functions while maintaining high accuracy in a large region of the (x,t) plane.