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Volume 32, Issue 3
Compact Difference Schemes for the Diffusion and Schrödinger Equations Approximation, Stability, Convergence, Effectiveness, Monotony

Vladimir A. Gordin & Eugeny A. Tsymbalov

DOI: 10.4208/jcm.1403-CR14

J. Comp. Math., 32 (2014), pp. 348-370.

Published online: 2014-06

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  • Abstract

Various compact difference schemes (both old and new, explicit and implicit, one-level and two-level), which approximate the diffusion equation and Schrödinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.

  • Keywords

Compact schemes, Pairs of test functions, Courant number, Two-level scheme, Order of convergence, Effectiveness, Monotony.

  • AMS Subject Headings

65M06.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-32-348, author = {Vladimir A. Gordin and Eugeny A. Tsymbalov}, title = {Compact Difference Schemes for the Diffusion and Schrödinger Equations Approximation, Stability, Convergence, Effectiveness, Monotony}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {3}, pages = {348--370}, abstract = {

Various compact difference schemes (both old and new, explicit and implicit, one-level and two-level), which approximate the diffusion equation and Schrödinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1403-CR14}, url = {http://global-sci.org/intro/article_detail/jcm/9891.html} }
TY - JOUR T1 - Compact Difference Schemes for the Diffusion and Schrödinger Equations Approximation, Stability, Convergence, Effectiveness, Monotony AU - Vladimir A. Gordin & Eugeny A. Tsymbalov JO - Journal of Computational Mathematics VL - 3 SP - 348 EP - 370 PY - 2014 DA - 2014/06 SN - 32 DO - http://doi.org/10.4208/jcm.1403-CR14 UR - https://global-sci.org/intro/article_detail/jcm/9891.html KW - Compact schemes, Pairs of test functions, Courant number, Two-level scheme, Order of convergence, Effectiveness, Monotony. AB -

Various compact difference schemes (both old and new, explicit and implicit, one-level and two-level), which approximate the diffusion equation and Schrödinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.

Vladimir A. Gordin and Eugeny A. Tsymbalov. (2014). Compact Difference Schemes for the Diffusion and Schrödinger Equations Approximation, Stability, Convergence, Effectiveness, Monotony. Journal of Computational Mathematics. 32 (3). 348-370. doi:10.4208/jcm.1403-CR14
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