Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model

Abdumauvlen Berdyshev, Dossan Baigereyev () and Kulzhamila Boranbek
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Abdumauvlen Berdyshev: Institute of Information and Computational Technologies, 28 Shevchenko Str., Almaty 050010, Kazakhstan
Dossan Baigereyev: Institute of Information and Computational Technologies, 28 Shevchenko Str., Almaty 050010, Kazakhstan
Kulzhamila Boranbek: Department of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, Kazakhstan

Mathematics, 2023, vol. 11, issue 17, 1-27

Abstract: This paper is aimed at efficient numerical implementation of the fractional-order generalization of the stochastic Stokes–Darcy model, which has important scientific, applied, and economic significance in hydrology, the oil industry, and biomedicine. The essence of this generalization of the stochastic model is the introduction of fractional time derivatives in the sense of Caputo’s definition to take into account long-term changes in the properties of media. An efficient numerical method for the implementation of the fractional-order Stokes–Darcy model is proposed, which is based on the use of a higher-order approximation formula for the fractional derivative, higher-order finite difference relations, and a finite element approximation of the problem in the spatial direction. In the paper, a rigorous theoretical analysis of the stability and convergence of the proposed numerical method is carried out, which is confirmed by numerous computational experiments. Further, the proposed method is applied to the implementation of the fractional-order stochastic Stokes–Darcy model using an ensemble technique, in which the approximation is carried out in such a way that the resulting systems of linear equations have the same coefficient matrix for all realizations. Furthermore, evaluation of the discrete fractional derivatives is carried out with the use of parallel threads. The efficiency of applying both approaches has been demonstrated in numerical tests.

Keywords: fractional-order stochastic Stokes–Darcy model; finite element method; convergence; parallel computing (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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