Abstracts

Novosibirsk participants

A class of implicit single-step algorithms is developed for solving the initial problem for system of ordinary differential equations. The algorithm is based on representation of the functions of the right-hand parts of equations by parametric polynomials. The polynomial coefficients are determined in the general case from (L+1) conditions of equality of the polynomials and their derivatives to the corresponding values of the approximated functions on the left at the section h (step of integration), (R+1) conditions on the right, and one condition of equality of the function values in the middle of the section. The local error at the step h is estimated; it has the order of ~h^Q+2, Q=(L+1)+(R+1) . The properties of A stability of the algorithms for R=L and, in addition, of L stability for R>L are proved. Approximate equations are obtained for variations, which are also solved using the proposed algorithm with initial conditions obtained from the solution of equations for variations at the previous step. This allows one to determine the integration error at the step h, use this information to adjust the step h in the course of integration, and evaluate the global accuracy.
If the function of the right-hand part depends only on the integration variable, these algorithms may be used for accurate calculation of certain integrals with a comparatively small number of intervals. Appropriate formulas are derived.
A simple modification of the method for R=L=1 is also developed, in which an additional condition of equality of the functions at the point (h-?),? << h is used. The method has the eighth order of accuracy in terms of h and possesses the properties of A-uL stability. Test examples are given.

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