An explicit method of the second order of accuracy for solving stiff systems of ordinary differential equations.

*(English. Russian original)*Zbl 0962.65056
Russ. Math. 42, No. 9, 52-60 (1998); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1998, No. 9, 55-63 (1998).

From the introduction: We consider an explicit numerical method for solving systems of ordinary differential equations. Explicit methods have both positive and negative features. Among positive ones we can cite the following: they are simple to realize, can be easily parallelized, and require relatively small memory.

In the papers of V. E. Lebedev [Sov. J. Numer. Anal. Math. Model. 4, No. 2, 111-135 (1989; Zbl 0825.65067); \((*)\) Numerical methods and applications 45-80 (1994; Zbl 0851.65052)], steps varied with respect to time are used, connected via certain relations with the roots of Chebyshev’s polynomial. This results in first-order schemes where the average step by time is higher than in schemes with a constant step. In \((*)\), methods of the second-order accuracy by time are also considered. The problem of construction of methods of that sort is reduced to construction of polynomials possessing properties of the Chebyshev alternate on a definite given segment and approximating the exponent with second-order accuracy in a neighborhood of zero. This problem can be reduced to a construction of polynomials with least deviation from zero with a weight. V. I. Lebedev [Russ. J. Numer. Anal. Math. Model. 9, No. 3, 231-263 (1994; Zbl 0818.65011)] has shown that many of these polynomials can be expressed via Zolotarov’s polynomials.

Applying these results, one can formulate the problem of construction of explicit methods of second-order with an enlarged stability domain and determine parameters of these methods, which are expressed via the roots of Zolotarov’s polynomials. The parameters of these polynomials are found. At the end of Section 4 we supply tables of roots; via these roots the parameters of an explicit method of second-order with maximal real stability domain are expressed.

In the papers of V. E. Lebedev [Sov. J. Numer. Anal. Math. Model. 4, No. 2, 111-135 (1989; Zbl 0825.65067); \((*)\) Numerical methods and applications 45-80 (1994; Zbl 0851.65052)], steps varied with respect to time are used, connected via certain relations with the roots of Chebyshev’s polynomial. This results in first-order schemes where the average step by time is higher than in schemes with a constant step. In \((*)\), methods of the second-order accuracy by time are also considered. The problem of construction of methods of that sort is reduced to construction of polynomials possessing properties of the Chebyshev alternate on a definite given segment and approximating the exponent with second-order accuracy in a neighborhood of zero. This problem can be reduced to a construction of polynomials with least deviation from zero with a weight. V. I. Lebedev [Russ. J. Numer. Anal. Math. Model. 9, No. 3, 231-263 (1994; Zbl 0818.65011)] has shown that many of these polynomials can be expressed via Zolotarov’s polynomials.

Applying these results, one can formulate the problem of construction of explicit methods of second-order with an enlarged stability domain and determine parameters of these methods, which are expressed via the roots of Zolotarov’s polynomials. The parameters of these polynomials are found. At the end of Section 4 we supply tables of roots; via these roots the parameters of an explicit method of second-order with maximal real stability domain are expressed.