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      Meshfree Semi-Lagrangian Methods for Solving Surface Advection PDEs

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          Abstract

          We analyze a class of meshfree semi-Lagrangian methods for solving advection problems on smooth, closed surfaces with solenoidal velocity field. In particular, we prove the existence of an embedding equation whose corresponding semi-Lagrangian methods yield the ones in the literature for solving problems on surfaces. Our analysis allows us to apply standard bulk domain convergence theories to the surface counterparts. In addition, we provide detailed descriptions for implementing the proposed methods to run on point clouds. After verifying the convergence rates against the theory, we show that the proposed method is a robust building block for more complicated problems, such as advection problems with non-solenoidal velocity field, inviscid Burgers’ equations and systems of reaction advection diffusion equations for pattern formation.

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          Total variation diminishing Runge-Kutta schemes

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            A simple embedding method for solving partial differential equations on surfaces

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              Semi-Lagrangian Methods for Level Set Equations

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                Author and article information

                Contributors
                argyrios.petras@ricam.oeaw.ac.at
                lling@hkbu.edu.hk
                sruuth@sfu.ca
                Journal
                J Sci Comput
                J Sci Comput
                Journal of Scientific Computing
                Springer US (New York )
                0885-7474
                1573-7691
                22 August 2022
                22 August 2022
                2022
                : 93
                : 1
                : 11
                Affiliations
                [1 ]GRID grid.475782.b, ISNI 0000 0001 2110 0463, Johann Radon Institute for Computational and Applied Mathematics (RICAM), ; Altenbergerstrasse 69, 4040 Linz, Austria
                [2 ]GRID grid.221309.b, ISNI 0000 0004 1764 5980, Hong Kong Baptist University, ; Kowloon Tong, Hong Kong
                [3 ]GRID grid.61971.38, ISNI 0000 0004 1936 7494, Simon Fraser University, ; Burnaby, Canada
                Author information
                http://orcid.org/0000-0002-3278-620X
                http://orcid.org/0000-0003-1779-0101
                http://orcid.org/0000-0002-9557-3375
                Article
                1966
                10.1007/s10915-022-01966-w
                9395508
                242e73c0-028a-4db0-b654-fa5706fa7df9
                © The Author(s) 2022

                Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

                History
                : 4 October 2021
                : 20 April 2022
                : 17 July 2022
                Funding
                Funded by: State of Upper Austria
                Funded by: FundRef http://dx.doi.org/10.13039/501100002790, Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada;
                Award ID: RGPIN 2016-04361
                Award Recipient :
                Funded by: Hong Kong Research Grant Council GRF
                Categories
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                Custom metadata
                © Springer Science+Business Media, LLC, part of Springer Nature 2022

                semi-lagrangian method,closest point method,radial basis functions,surface conservation laws,65m06,65m25

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