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The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive manner, as well as ensuring that the discrete conformation tensor is positive definite. A full implementation with coupling to the fluid flow is presented, along with a detailed discussion of the issues that arise with such schemes. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.<\/jats:p>","DOI":"10.1007\/s10444-024-10211-x","type":"journal-article","created":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T06:03:21Z","timestamp":1734415401000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Discretisation of an Oldroyd-B viscoelastic fluid flow using a Lie derivative formulation"],"prefix":"10.1007","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-3628-4052","authenticated-orcid":false,"given":"Ben S.","family":"Ashby","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4499-0563","authenticated-orcid":false,"given":"Tristan","family":"Pryer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,12,17]]},"reference":[{"key":"10211_CR1","doi-asserted-by":"crossref","unstructured":"Aidun, C. 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