Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model
Baishemirov Z. Berdyshev A. Baigereyev D. Boranbek K.
August 2024Multidisciplinary Digital Publishing Institute (MDPI)
Fractal and Fractional
2024#8Issue 8
This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions.
Caputo fractional derivative , convergence rate , ensemble method , numerical method , sparse grid stochastic collocation method , Stokes–Darcy equations
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Department of Mathematics and Mathematical Modeling, Abai Kazakh National Pedagogical University, 13, Dostyk Ave., Almaty, 050010, Kazakhstan
Department of Science, Abai Kazakh National Pedagogical University, 13, Dostyk Ave., Almaty, 050010, Kazakhstan
Institute of Information and Computational Technologies, 28, Shevchenko Str., Almaty, 050010, Kazakhstan
School of Applied Mathematics, Kazakh-British Technical University, 59, Tole bi Str., Almaty, 050005, Kazakhstan
Department of Mathematics, Sarsen Amanzholov East Kazakhstan University, 148, Shakarim Ave., Ust-Kamenogorsk, 070002, Kazakhstan
Department of Mathematics and Mathematical Modeling
Department of Science
Institute of Information and Computational Technologies
School of Applied Mathematics
Department of Mathematics
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