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Article information
2015 , Volume 20, ¹ 2, p.56-64
Paasonen V.I.
Compact third-order accuracy schemes on non-uniform adaptive grids
Compact difference schemes of the third order of accuracy (with the second order of approximation on an evolutionary variable) on non-uniform grids are created for both one-dimensional nonlinear Schrodinger equation and the heat conductivity equation. The purpose of this study aims at the development of such difference method for the considered equations in which some approaches for the increase of the efficiency of calculations are combined. These approaches include a non-uniform grid, a high order of approximation, lack of iterations on nonlinearity, dynamic adaptation of the grid to the solution and rather exact interpolation of the solution on a new grid. Accurate calculation of solutions with the concentrated waves in case of uniform grids requires quite small step. However, the small step is not required in zones with moderate values of gradient. Therefore, in this research we prefer to use the non-uniform grids. Another useful tool for improvement of the quality of calculations that we apply is the higher order of accuracy. For this purpose we implement a generalization of known compact schemes on a case of non-uniform grids in this research. The high-accurate technology of calculation is supplemented with the mechanism of adaptation of a grid to the solution. For this purpose, the evolutionary variable is reconstructed on each step, and the solution is interpolated on the new grid with sufficient accuracy. Formally, here we apply the two-layer schemes with double step on three layers. The center layer is used only for the approximation of non-linear part in the right hand side of equations. It allows us to exclude the need for iterations. Results of comparative calculations for compact schemes and known usual schemes are given in the final part of research. All carried-out calculations have shown the essential advantage of compact schemes in the accuracy and economy of resources.
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Keywords: Shrodinger equation, compact difference scheme, scheme on nonuniform grids, high-order accuracy scheme, nonlinear fiber optics
Author(s):
Paasonen Viktor Ivanovich
PhD. , Associate Professor
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave. 6
Phone Office: (383) 330 86 56
E-mail: paas@ict.nsc.ru
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Bibliography link:
Paasonen V.I. Compact third-order accuracy schemes on non-uniform adaptive grids // Computational technologies. 2015. V. 20. ¹ 2. P. 56-64
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