Numerical Methods of Solving Cauchy Problems with Contrast Structures
https://doi.org/10.18255/1818-1015-2016-5-529-538
Abstract
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.
About the Authors
A. A. Belov
Lomonosov Moscow State University, 1–2 Leninskie Gory, Moscow 119991, Russia
Russian Federation
graduate student
Russian Federation
graduate student
N. N. Kalitkin
Keldysh Institute of Applied Mathematics RAS, 4 Miusskaya sq., Moscow 125047, Russia
Russian Federation
corresponding member RAS, Dr. Sci. (Phys.–Math.), professor
Russian Federation
corresponding member RAS, Dr. Sci. (Phys.–Math.), professor
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Review
For citations:
Belov A.A., Kalitkin N.N. Numerical Methods of Solving Cauchy Problems with Contrast Structures. Modeling and Analysis of Information Systems. 2016;23(5):529-538. (In Russ.) https://doi.org/10.18255/1818-1015-2016-5-529-538
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