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Vestnik KRAUNC. Fiziko-Matematicheskie Nauki, 2024, Volume 46, Number 1, Pages 103–117
DOI: https://doi.org/10.26117/2079-6641-2024-46-1-103-117
(Mi vkam639)
 

INFORMATION AND COMPUTATION TECHNOLOGIES

Application of high-performance computing to solve the cauchy problem with the fractional Riccati equation using an nonlocal implicit finite-difference scheme

D. A. Tvyordyj, R. I. Parovik

Institute for Cosmophysical Research and Radio Wave Propagation FEB RAS
References:
Abstract: The article presents a study of the computational efficiency of a parallel version of a numerical algorithm for solving the Riccati equation with a fractional variable order derivative of the Gerasimov-Caputo type. The numerical algorithm is a nonlocal implicit finite-difference scheme, which reduces to a system of nonlinear algebraic equations and is solved using a modified Newton method. The nonlocality of the numerical scheme creates a high computational load on computing resources, which creates the need to implement efficient parallel algorithms for solving them. The numerical algorithm studied for efficiency is implemented in the C language due to its versatility when working with memory. Parallelization was carried out using OpenMP technology. A series of computational experiments are being carried out on the NVIDIA DGX STATION computing server (Institute of Mathematics named after V.I. Romanovsky, Tashkent, Uzbekistan) and the HP Pavilion Gaming Laptop Z270X, where the Cauchy problem for the fractional Riccati equation with non-constant coefficients was solved. Based on the average computation time, the speedup, efficiency and cost of the algorithm are calculated. From the data analysis it is clear that the OpenMP parallel software implementation of the non-local implicit finite-difference scheme shows an acceleration of 9-12 times, depending on the number of CPU cores involved.
Keywords: parallel computing, OpenMP, implicit finite difference schemes, Newton's method, fractional derivatives, memory effect, non-locality, non-linearity.
Funding agency Grant number
Russian Science Foundation 22-11-00064
The name of the funding programme: The research was carried out within the framework of the RSF grant № 22-11-00064 on the topic “Modelling of dynamic processes in geospheres taking into account heredity”(https://rscf.ru/project/22-11-00064/). Organization that has provided funding: Russian Scientific Foundation.
Document Type: Article
UDC: 519.642.2, 519.687.1
MSC: Primary 34A08; Secondary 65Y05, 65M06
Language: Russian
Citation: D. A. Tvyordyj, R. I. Parovik, “Application of high-performance computing to solve the cauchy problem with the fractional Riccati equation using an nonlocal implicit finite-difference scheme”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 46:1 (2024), 103–117
Citation in format AMSBIB
\Bibitem{TvyPar24}
\by D.~A.~Tvyordyj, R.~I.~Parovik
\paper Application of high-performance computing to solve the cauchy problem with the fractional Riccati equation using an nonlocal implicit finite-difference scheme
\jour Vestnik KRAUNC. Fiz.-Mat. Nauki
\yr 2024
\vol 46
\issue 1
\pages 103--117
\mathnet{http://mi.mathnet.ru/vkam639}
\crossref{https://doi.org/10.26117/2079-6641-2024-46-1-103-117}
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