Geodesic
The special difference schemes for numerical solution of ordinary differential equation with the alternating-sign coefficient
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2010), pp. 5-17

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The special difference first and second order schemes for the through numerical calculation of the Cauchy problem for an ordinary differential equation with the alternating-sign coefficient at the unknown function are proposed. The construction of schemes is based on the use of the exact integral solution of the equation and approximations of the grid functions invariant to the sign of the equation's coefficient. Properties, asymptotics, and the rational approximation of the schemes are considered. An opportunity of application of the schemes at rough steps of integration and convergence of numerical results to exact solutions is shown on test examples.
Keywords: ordinary differential equation, first order, alternating-sign coefficient, special difference schemes, through numerical calculation. 
@article{VTGU_2010_4_a0,
     author = {V. G. Zverev},
     title = {The special difference schemes for numerical solution of ordinary differential equation with the alternating-sign coefficient},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {5--17},
     publisher = {mathdoc},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2010_4_a0/}
}
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V. G. Zverev. The special difference schemes for numerical solution of ordinary differential equation with the alternating-sign coefficient. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2010), pp. 5-17. http://geodesic.mathdoc.fr/item/VTGU_2010_4_a0/