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Голуба - Кахана. Описываются алгоритмы с односторонним и двусторонним предобусловливанием СЛАУ. Исследуются обобщения метода бидиагонализации Голуба - Кахана - Ариоли для решения алгебраических систем с матрицами седлового типа, возникающими в актуальных задачах электромагнетизма, фильтрации и других приложениях. Обсуждаются вопросы масштабируемого распараллеливания средствами гибридного программирования на многопроцессорных вычислительных системах с распределенной и иерархической общей памятью.
The stability properties of symmetric multi-step methods in the predictor-corrector mode
E. D. Karepova, Yu. V. Shan'ko, I. R. Adaev Institute of Computational Modeling SB RAS Email: e.d.karepova@icm.krasn.ru DOI: 10.24412/cl-35065-2022-1-00-19
The orbital motion is described by the system of second-order ordinary differential equations. J. Lambert and I. Watson proposed the symmetric methods [1], that possess a periodicity property when the product of the step-size and the angular frequency lies within a certain interval called the interval of periodicity. The numerical integration of orbit by the symmetric methods with the step-size from the interval of periodicity gives the longitude error which increases linearly, whereas the energy error remains roughly constant, which distinguishes symmetric methods from other one for the better.
The symmetric methods are not uniquely determined even if their order and explicitness are specified. We construct and investigate the high-order symmetric explicit and implicit methods in the "Predict-Evaluate-Cor-rect-Evaluate" (PECE) mode. We propose a technology for testing the stability of predictor-corrector pairs and obtaining their intervals of periodicity or stability. Note that the symmetry is preserved only in the case when the order of the corrector is one greater than the order of the predictor, otherwise the pair should be examined for stability.
This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2022-873).
References
1. J. D. Lambert, I. A. Watson. Symmetric Multistep Methods for Periodic Initial Value Problems // J. Inst. Maths Applics. 1976. V. 18. P. 189-202.
On convergence of shock-capturing schemes inside the shocks influence area
O. A. Kovyrkina12, V. V. Ostapenko12, V. F. Tishkin3
1Lavrentyev Institute of Hydrodynamics SB RAS
2Novosibirsk State University
3Keldysh Institute of Applied Mathematics RAS
Email: olyana@ngs.ru
DOI: 10.24412/cl-35065-2022-1-00-20
We show that in the Nonlinear Flux Correction (NFC) schemes CABARETM [1] and WENO5 [2] (in contrast to the Rusanov scheme [3]), there is no local convergence of the difference solution inside the shock influence area when calculating the shallow water dam break problem. This is due to the fact that the numerical solutions obtained by these schemes have persistent oscillations inside the domain of a constant flow between a shock and a centered rarefaction wave. In this case, taking into account the Lax - Wendroff theorem [4], the numerical solutions obtained by the NFC schemes converge to the exact solution inside the shock influence area in the weak sense only.
The reported study was funded by RFBR and NSFC, project number 21-51-53012.
