Second-Order Finite Difference Approximations of the Upper-Convected Time Derivative
Abstract
In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation of the popular viscoelastic models, is reformulated in order to obtain approximations of second-order in time for solving a simplified constitutive equation in one and two dimensions. The theoretical analysis of the truncation errors of the methods takes into account the linear and quadratic interpolation operators based on a Lagrangian framework. Numerical experiments illustrating the theoretical results for the model equation defined in one and two dimensions are included. Finally, the finite difference approximations of second-order in time are also applied for solving a two-dimensional Oldroyd-B constitutive equation subjected to a prescribed velocity field at different Weissenberg numbers.
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© 2021, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License
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Submitted: 8 September 2020
Accepted: 10 September 2021
Published online: 2 December 2021
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Center of Mathematical Sciences Applied to Industry : 2013/07375-0
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Fundação de Amparo à Pesquisa do Estado de São Paulo https://doi.org/10.13039/501100001807 : 2013/07375-0
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Fundação de Amparo à Pesquisa do Estado de São Paulo https://doi.org/10.13039/501100001807 : 2019/08742-2, 2017/11428-2
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Conselho Nacional de Desenvolvimento Científico e Tecnológico https://doi.org/10.13039/501100003593 : 305383/2019-1
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Japan Science and Technology Agency https://doi.org/10.13039/501100002241 : JP-MJPR16EA, JPMJCR2014
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Japan Society for the Promotion of Science https://doi.org/10.13039/501100001691 : JP18H01135, JP20H01823, JP20KK0058, JP21H04431
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