DOI DOI 10.15507/2079-6900.28.202602.99-114
Original article
ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)
MSC2020 76-10
Comparative analysis of linear solvers for the Jacobian-free Newton–Krylov method for solving gas dynamics problems using an implicit scheme for the Discontinuous Galerkin Method
Z. R. Viktorovich, M. A. Dmitrievich, V. V. Vladimirovich
National Research Mordovia State University (Saransk, Russian Federation)
Abstract. A comparative study of the efficiency of linear solvers within the Jacobian-free Newton--Krylov (JFNK) framework for the numerical simulation of two-dimensional gas dynamics equations using the discontinuous Galerkin method is presented. The classical MINRES method and the modern induced dimension reduction method IDR(s) with various values of parameter $s$ are considered. The study is performed on the example of the Kelvin—Helmholtz instability problem. The accuracy of the obtained numerical solutions, spectral characteristics of the flow, and computational performance of the algorithms are analyzed. All IDR(s) variants are shown to provide a quality of vortex structure resolution and energy spectra comparable to that of MINRES. Meanwhile, all methods from the IDR(s) family require significantly less wall-clock time per time step compared to MINRES, achieving a speedup factor from 1.5 to 2.5, depending on the dimension of the auxiliary subspace. The IDR(4) method demonstrates the best balance between convergence rate and computational cost, which allows us to recommend it as an efficient linear solver within the JFNK approach for unsteady computational fluid dynamics problems.
Key Words: Discontinuous Galerkin Method, Implicit Scheme, Jacobian–free Newton– Krylov Method, MINRES method, IDR(s) method
For citation: Z. R. Viktorovich, M. A. Dmitrievich, V. V. Vladimirovich. Comparative analysis of linear solvers for the Jacobian-free Newton–Krylov method for solving gas dynamics problems using an implicit scheme for the Discontinuous Galerkin Method. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:2(2026), 99–114. DOI: https://doi.org/DOI 10.15507/2079-6900.28.202602.99-114
Submitted: 01.12.2021; Revised: 10.02.2022; Accepted: 24.02.2022
Information about the authors:
Zhalnin Ruslan. Viktorovich, Dean of the Faculty of Mathematics and IT, Ogarev Mordovia State University (68/1 Bolshevistskaya St., Saransk, 430005, Russia), ORCID: https://orcid.org/0000-0002-1103-3321, zhalninrvrv@yandex.ru
Mulyugin Alexander. Dmitrievich, Laboratory research assistant, Faculty of Mathematics and IT, Ogarev Mordovia State University (68/1 Bolshevistskaya St., Saransk, 430005, Russia), ORCID: https://orcid.org/0009-0004-2721-1390, alexandermulyugin@yandex.ru
Vdovin Vladislav. Vladimirovich, Laboratory research assistant, Faculty of Mathematics and IT, Ogarev Mordovia State University (68/1 Bolshevistskaya St., Saransk, 430005, Russia), ORCID: https://orcid.org/0009-0002-2558-4324, vdovinvv05@mail.ru
All authors have read and approved the final manuscript.
Conflict of interest: The authors declare no conflict of interest.
This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License. 