Efficient Mapping of High Order Basis Sets for Unbounded Domains (Communications in Computational Physics, 2018)
Many physical problems involve unbounded domains where the physicalquantities vanish at infinities. Numerically, this has been handled using different techniquessuch as domain truncation, approximations using infinitely extended and vanishingbasis sets, and mapping bounded basis sets using some coordinate transformations.Each technique has its own advantages and disadvantages. Yet, approximatingsimultaneously and efficiently a wide range of decaying rates has persisted as majorchallenge. Also, coordinate transformation, if not carefully implemented, can result innon-orthogonal mapped basis sets. In this work, we revisited this issue with an emphasizeon designing appropriate transformations using sine series as basis set. Thetransformations maintain both the orthogonality and the efficiency. Furthermore, usingsimple basis set (sine function) help avoid the expensive numerical integrations. Inthe calculations, four types of physically recurring decaying behaviors are considered,which are: non-oscillating and oscillating exponential decays, and non-oscillating andoscillating algebraic decays. The results and the analyses show that properly designedhigh-order mapped basis sets can be efficient tools to handle challenging physicalproblems on unbounded domains. Decay rate ranges as large of 6 orders of magnitudescan be approximated efficiently and concurrently.