A High-Order Finite Element Method for Solving Two-Dimensional Fractional Rayleigh–Stokes Problem for a Heated Generalized Second Grade Fluid

IF 1.7 4区工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

International Journal for Numerical Methods in Fluids Pub Date : 2024-12-30 DOI:10.1002/fld.5361

Eric Ngondiep

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Abstract

This article develops a high-order finite element scheme in an approximate solution of the two-dimensional Rayleigh–Stokes problem for a heated generalized second-grade fluid with fractional derivatives. The constructed approach consists of approximating the exact solution by interpolation in time while the finite element technique is used in the approximation of the spatial derivatives. This combination is simple and easy to implement. The stability and error estimates of the developed strategy are deeply analyzed in the $L^{\infty}$ -norm. The theoretical studies suggest that the proposed method is unconditionally stable, convergent with order $O (σ^{1 + γ} + h^{p})$ , faster, and more efficient than a broad range of numerical schemes discussed in the literature for the considered time fractional partial differential equation. Some numerical examples are carried out to show the applicability and viability of the new algorithm.

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求解二阶加热广义二阶流体二维分数阶Rayleigh-Stokes问题的高阶有限元方法

本文发展了含分数阶导数的加热广义二阶流体二维瑞利-斯托克斯问题近似解的高阶有限元格式。所构造的方法是在时间上用插值逼近精确解，而在空间导数上用有限元技术逼近。这种组合简单且易于实现。在L∞$$ {L}^{\infty } $$ -范数下，对所开发策略的稳定性和误差估计进行了深入分析。理论研究表明，该方法是无条件稳定的；收敛于O （σ 1 + γ + h p）阶) $$ O\left({\sigma}^{1+\gamma }+{h}^p\right) $$，比文献中讨论的考虑时间分数阶偏微分方程的广泛数值格式更快，更有效。算例表明了新算法的适用性和可行性。

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来源期刊

International Journal for Numerical Methods in Fluids 物理-计算机：跨学科应用

CiteScore

3.70

自引率

5.60%

发文量

111

审稿时长

8 months

期刊介绍： The International Journal for Numerical Methods in Fluids publishes refereed papers describing significant developments in computational methods that are applicable to scientific and engineering problems in fluid mechanics, fluid dynamics, micro and bio fluidics, and fluid-structure interaction. Numerical methods for solving ancillary equations, such as transport and advection and diffusion, are also relevant. The Editors encourage contributions in the areas of multi-physics, multi-disciplinary and multi-scale problems involving fluid subsystems, verification and validation, uncertainty quantification, and model reduction. Numerical examples that illustrate the described methods or their accuracy are in general expected. Discussions of papers already in print are also considered. However, papers dealing strictly with applications of existing methods or dealing with areas of research that are not deemed to be cutting edge by the Editors will not be considered for review. The journal publishes full-length papers, which should normally be less than 25 journal pages in length. Two-part papers are discouraged unless considered necessary by the Editors.