Journal Pre-proof A meshless technique based on generalized moving least squares combined with the second-order semi-implicit backward differential formula for numerically solving time-dependent phase ﬁeld models on the spheres Vahid Mohammadi, Mehdi Dehghan PII: S0168-9274(20)30058-1 DOI: https://doi.org/10.1016/j.apnum.2020.02.012 Reference: APNUM 3777 To appear in: Applied Numerical Mathematics Received date: 29 August 2019 Revised date: 18 February 2020 Accepted date: 18 February 2020 Please cite this article as: V. Mohammadi, M. Dehghan, A meshless technique based on generalized moving least squares combined with the second-order semi-implicit backward differential formula for numerically solving time-dependent phase ﬁeld models on the spheres, Appl. Numer. Math. (2020), doi: https://doi.org/10.1016/j.apnum.2020.02.012. This is a PDF ﬁle of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the deﬁnitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its ﬁnal form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.