The numerical unified transform method for initial-boundary value problems on the half-line. (4th March 2021)

• Record Type: Journal Article
• Authors: Deconinck, Bernard; Trogdon, Thomas; Yang, Xin
• Is Part Of: IMA journal of numerical analysis. Volume 42:Number 2(2022)
• Page Start: 1400
• Page End: 1433
• Publication Date: 2021-03-04
• Subjects: linear evolution partial differential equations -- numerical unified transform method -- method of steepest descent -- numerical oscillatory integrals; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drab007
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: We implement the unified transform method of Fokas as a numerical method to solve linear evolution 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 (absolute errors are smaller, relative errors are bounded). Our goal is to make no assumptions on the functional form of the initial or boundary functions beyond some decay and smoothness, while maintaining high accuracy in a large region of the $(x, t)$ plane.