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Eulerian\u2013Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine\u2013Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for the mesh cells to handle intersecting characteristics due to shocks. Following this partitioning, we write the equation in a time-differential form and evolve with Runge\u2013Kutta methods in a method-of-lines fashion. High-resolution methods such as ENO and WENO-AO schemes are used for spatial reconstruction. Extension to higher dimensions is done via dimensional splitting. Numerical experiments demonstrate our scheme\u2019s high-order accuracy and ability to sharply capture post-shock solutions with large time-stepping sizes.<\/jats:p>","DOI":"10.1007\/s10915-024-02714-y","type":"journal-article","created":{"date-parts":[[2024,11,18]],"date-time":"2024-11-18T18:29:37Z","timestamp":1731954577000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A High-Order Eulerian\u2013Lagrangian Runge\u2013Kutta Finite Volume (EL\u2013RK\u2013FV) Method for Scalar Nonlinear Conservation Laws"],"prefix":"10.1007","volume":"102","author":[{"given":"Jiajie","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-6589-4013","authenticated-orcid":false,"given":"Joseph","family":"Nakao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jing-Mei","family":"Qiu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,11,18]]},"reference":[{"key":"2714_CR1","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1016\/j.matcom.2016.12.012","volume":"137","author":"E Abreu","year":"2017","unstructured":"Abreu, E., et al.: A new finite volume approach for transport models and related applications with balancing source terms. 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