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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Main Authors: A. A. Belov, N. N. Kalitkin
Format: Article
Language:English
Published: Yaroslavl State University 2016-10-01
Series:Моделирование и анализ информационных систем
Subjects:
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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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