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The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers\u2019 equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers\u2019 equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers\u2019 equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.<\/jats:p>","DOI":"10.1007\/s10915-022-02023-2","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T22:02:38Z","timestamp":1668031358000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds"],"prefix":"10.1007","volume":"93","author":[{"given":"Yinghua","family":"Wang","sequence":"first","affiliation":[]},{"given":"Bao-Shan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Leevan","family":"Ling","sequence":"additional","affiliation":[]},{"given":"Wai Sun","family":"Don","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,9]]},"reference":[{"issue":"6","key":"2023_CR1","doi-asserted-by":"publisher","first-page":"1909","DOI":"10.1111\/j.1467-8659.2012.03071.x","volume":"31","author":"S Auer","year":"2012","unstructured":"Auer, S., Macdonald, C.B., Treib, M., Schneider, J., Westermann, R.: Real-time fluid effects on surfaces using the closest point method. 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