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Discretization Errors of Perturbation Numerical Methods for Initial Value Problems in Ordinary Differential Equations

机译：常微分方程初值问题扰动数值方法的离散化误差

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Algorithms are discussed, which are obtained within the perturbation numerical (PN) approach to the solution of initial value problems for linear ordinary differential equations and maxnorm bounds are derived to their accumulated discretization errors. The importance is emphasized of the realization of piecewise polynomial approximations for the coefficients of the differential equation by means of truncated local Legendre series. On one hand, this is proved to ensure superconvergence of the zeroth order approximation of perturbation theory with minimum smoothness requirements on the original coefficients. On the other hand, convenient cut offs of the Legendre series can be defined which ensure the implementation, in the p-th order approximation of perturbation theory (p=1,2,...) of finite PN algorithms that preserve the highest theoretically attainable order of accuracy at given p. The error analysis also shows that, beyond the zeroth order approximation of perturbation theory, an increased efficiency of the PN methods is got provided the differential equation to be solved is first brought to the normal form. (Atomindex citation 16:019744)

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