{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/171106212","title":"Fifth order finite volume WENO in general orthogonally-curvilinear coordinates","arxiv_id":"1711.06212","date":"2017-11-16","proceeding":null,"authors":["Mohammad Afzal Shadab","Dinshaw Balsara","Wei Shyy","Kun Xu"],"abstract":"High order reconstruction in the finite volume (FV) approach is achieved by a more fundamental form of the fifth order WENO reconstruction in the framework of orthogonally-curvilinear coordinates, for solving the hyperbolic conservation equations. The derivation employs a piecewise parabolic polynomial approximation to the zone averaged values to reconstruct the right, middle, and left interface values. The grid dependent linear weights of the WENO are recovered by inverting a Vandermode-like linear system of equations with spatially varying coefficients. A scheme for calculating the linear weights, optimal weights, and smoothness indicator on a regularly- and irregularly-spaced grid in orthogonally-curvilinear coordinates is proposed. A grid independent relation for evaluating the smoothness indicator is derived from the basic definition. Finally, the procedures for the source term integration and extension to multi-dimensions are proposed. Analytical values of the linear and optimal weights, and also the weights required for the source term integration and flux averaging, are provided for a regularly-spaced grid in Cartesian, cylindrical, and spherical coordinates. Conventional fifth order WENO reconstruction for the regularly-spaced grids in the Cartesian coordinates can be fully recovered in the case of limiting curvature. The fifth order finite volume WENO-C (orthogonally-curvilinear version of WENO) reconstruction scheme is tested for several 1D and 2D benchmark test cases involving smooth and discontinuous flows in cylindrical and spherical coordinates.","url_abs":"http://arxiv.org/abs/1711.06212v2","url_pdf":"http://arxiv.org/pdf/1711.06212v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"171106212","repo_url":"https://github.com/mashadab/WENO-curvilinear","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}