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A reduced‐order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain
Author(s) -
Teng Fei,
Luo Zhengdong,
Yang Jing
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5647
Subject(s) - mathematics , discretization , convergence (economics) , hyperbolic partial differential equation , mathematical analysis , stability (learning theory) , domain (mathematical analysis) , boundary (topology) , boundary value problem , finite element method , partial differential equation , computer science , physics , machine learning , economics , thermodynamics , economic growth
In this article, we primarily focuses to study the order‐reduction for the classical natural boundary element (NBE) method for the two‐dimensional (2D) hyperbolic equation in unbounded domain. To this end, we first build a semi‐discretized format about time for the hyperbolic equation and discuss the existence, stability, and convergence of the time semi‐discretized solutions. We then establish the classical fully discretized NBE format from the time semi‐discretized one and analyze the existence, stability, and convergence of the classical NBE solutions. Next, using proper orthogonal decomposition method, we build a reduced‐order extrapolated NBE (ROENBE) format containing very few unknowns but having adequately high accuracy, and we also discuss the existence, stability, and convergence of the ROENBE solutions. Finally, we use some numerical examples to show that the ROENBE method is far superior to the classical NBE one. It shows that the ROENBE method is reliable and effective for solving the 2D hyperbolic equation with the unbounded domain.