Numerical Methods of Solving Cauchy Problems with Contrast Structures
Modern numerical methods allowing to solve contrast structure problems in the most efficient way are described. These methods include explicit-implicit Rosenbrock schemes with complex coefficients and fully implicit backward optimal Runge–Kutta schemes. As an integration argument, it is recommended...
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Format: | Article |
Language: | English |
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Yaroslavl State University
2016-10-01
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Series: | Моделирование и анализ информационных систем |
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Online Access: | https://www.mais-journal.ru/jour/article/view/386 |
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author | A. A. Belov N. N. Kalitkin |
author_facet | A. A. Belov N. N. Kalitkin |
author_sort | A. A. Belov |
collection | DOAJ |
description | Modern numerical methods allowing to solve contrast structure problems in the most efficient way are described. These methods include explicit-implicit Rosenbrock schemes with complex coefficients and fully implicit backward optimal Runge–Kutta schemes. As an integration argument, it is recommended to choose the length of the integral curve arc. This argument provides high reliability of the calculation and sufficiently decreases the complexity of computations for low-order systems. In order to increase the efficiency, we propose an automatic step selection algorithm based on curvature of the integral curve. This algorithm is as efficient as standard algorithms and has sufficiently larger reliability. We show that along with such an automatic step selection it is possible to calculate a posteriori asymptotically precise error estimation. Standard algorithms do not provide such estimations and their actual error quite often exceeds the user-defined tolerance by several orders. The applicability limitations of numerical methods are investigated. In solving superstiff problems, they sometimes do not provide satisfactory results. In such cases, it is recommended to imply approximate analytical methods. Consequently, numerical and analytical methods are complementary. |
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institution | Directory Open Access Journal |
issn | 1818-1015 2313-5417 |
language | English |
last_indexed | 2025-03-14T08:52:41Z |
publishDate | 2016-10-01 |
publisher | Yaroslavl State University |
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series | Моделирование и анализ информационных систем |
spelling | doaj.art-2728bce772eb4fb98439d255c810132d2025-03-02T12:46:58ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-10-0123552953810.18255/1818-1015-2016-5-529-538322Numerical Methods of Solving Cauchy Problems with Contrast StructuresA. A. Belov0N. N. Kalitkin1Lomonosov Moscow State University, 1–2 Leninskie Gory, Moscow 119991, RussiaKeldysh Institute of Applied Mathematics RAS, 4 Miusskaya sq., Moscow 125047, RussiaModern numerical methods allowing to solve contrast structure problems in the most efficient way are described. These methods include explicit-implicit Rosenbrock schemes with complex coefficients and fully implicit backward optimal Runge–Kutta schemes. As an integration argument, it is recommended to choose the length of the integral curve arc. This argument provides high reliability of the calculation and sufficiently decreases the complexity of computations for low-order systems. In order to increase the efficiency, we propose an automatic step selection algorithm based on curvature of the integral curve. This algorithm is as efficient as standard algorithms and has sufficiently larger reliability. We show that along with such an automatic step selection it is possible to calculate a posteriori asymptotically precise error estimation. Standard algorithms do not provide such estimations and their actual error quite often exceeds the user-defined tolerance by several orders. The applicability limitations of numerical methods are investigated. In solving superstiff problems, they sometimes do not provide satisfactory results. In such cases, it is recommended to imply approximate analytical methods. Consequently, numerical and analytical methods are complementary.https://www.mais-journal.ru/jour/article/view/386stiff cauchy problemcontrast structureautomatic step selectioncurvature in multidimensional spacerichardson method estimationssingularity diagnosticssolution blow-up |
spellingShingle | A. A. Belov N. N. Kalitkin Numerical Methods of Solving Cauchy Problems with Contrast Structures Моделирование и анализ информационных систем stiff cauchy problem contrast structure automatic step selection curvature in multidimensional space richardson method estimations singularity diagnostics solution blow-up |
title | Numerical Methods of Solving Cauchy Problems with Contrast Structures |
title_full | Numerical Methods of Solving Cauchy Problems with Contrast Structures |
title_fullStr | Numerical Methods of Solving Cauchy Problems with Contrast Structures |
title_full_unstemmed | Numerical Methods of Solving Cauchy Problems with Contrast Structures |
title_short | Numerical Methods of Solving Cauchy Problems with Contrast Structures |
title_sort | numerical methods of solving cauchy problems with contrast structures |
topic | stiff cauchy problem contrast structure automatic step selection curvature in multidimensional space richardson method estimations singularity diagnostics solution blow-up |
url | https://www.mais-journal.ru/jour/article/view/386 |
work_keys_str_mv | AT aabelov numericalmethodsofsolvingcauchyproblemswithcontraststructures AT nnkalitkin numericalmethodsofsolvingcauchyproblemswithcontraststructures |