This article is cited in 1 scientific paper (total in 1 paper)
Numerical methods of solving Cauchy problems with contrast structures
A. A. Belova, N. N. Kalitkinb
a Lomonosov Moscow State University,
1–2 Leninskie Gory, Moscow 119991, Russia
b Keldysh Institute of Applied Mathematics RAS,
4 Miusskaya sq., Moscow 125047, Russia
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.
stiff Cauchy problem, contrast structure, automatic step selection, curvature in multidimensional space, Richardson method estimations, singularity diagnostics, solution blow-up.
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A. A. Belov, N. N. Kalitkin, “Numerical methods of solving Cauchy problems with contrast structures”, Model. Anal. Inform. Sist., 23:5 (2016), 529–538
Citation in format AMSBIB
\by A.~A.~Belov, N.~N.~Kalitkin
\paper Numerical methods of solving Cauchy problems with contrast structures
\jour Model. Anal. Inform. Sist.
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N. N. Kalitkin, A. A. Belov, P. E. Bulatov, “Raschet khimicheskoi kinetiki yavnymi skhemami s geometricheski-adaptivnym vyborom shaga”, Preprinty IPM im. M. V. Keldysha, 2018, 173, 32 pp.
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