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Geometrically adaptive grids for stiff Cauchy problems. (English. Russian original) Zbl 1341.65026

Dokl. Math. 93, No. 1, 112-116 (2016); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 466, No. 3, 276-281 (2016).
Summary: A new method for automatic step size selection in the numerical integration of the Cauchy problem for ordinary differential equations is proposed. The method makes use of geometric characteristics (curvature and slope) of an integral curve. For grids generated by this method, a mesh refinement procedure is developed that makes it possible to apply the Richardson method and to obtain a posteriori asymptotically precise estimate for the error of the resulting solution (no such estimates are available for traditional step size selection algorithms). Accordingly, the proposed methods are more robust and accurate than previously known algorithms. They are especially efficient when applied to highly stiff problems, which is illustrated by numerical examples.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L04 Numerical methods for stiff equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations

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References:

[1] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer-Verlag, Berlin, 1996; Mir, Moscow, 1999). · Zbl 0859.65067
[2] Vasil’eva, A. B.; Butuzov, V. F.; Nefedov, N. N., No article title, Fundam. Prikl. Mat., 4, 799-851 (1998) · Zbl 0963.34043
[3] N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov, Computations on Quasi-Uniform Meshes (Fizmatlit, Moscow, 2005) [in Russian].
[4] Kalitkin, N. N.; Poshivaylo, I. P., No article title, Dokl. Math., 85, 139-143 (2012) · Zbl 1238.65070 · doi:10.1134/S1064562412010103
[5] Kalitkin, N. N.; Poshivaylo, I. P., No article title, Math. Model. Comput. Simul., 6, 272-285 (2014) · Zbl 1356.65176 · doi:10.1134/S2070048214030077
[6] I. P. Poshivaylo, Candidate’s Dissertation in Mathematics and Physics (Moscow, 2015).
[7] Shampine, L. F.; Reichelt, M. W., No article title, SIAM J. Sci. Comput., 18, 1-22 (1997) · Zbl 0868.65040 · doi:10.1137/S1064827594276424
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