Geodesic
Linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients, obtained by minimizing the approximation error
Čebyševskij sbornik, Tome 21 (2020) no. 4, pp. 243-256

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The work is devoted to determining the range of grid Peclet number values, for which the proposed scheme, representing a linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients, obtained by minimizing the approximation error, has higher accuracy than conventional schemes, including modifications Upwind Leapfrog difference scheme with limiters. The article obtained a limit on the time step for a difference scheme with weights at which the calculation error is in an acceptable range. It is shown that the proposed scheme, based on the basis of a linear combination of the Upwind and Standard Leapfrog difference schemes with weighting coefficients $2/3$ and $1/3$, respectively, obtained by minimizing the approximation error more precisely, the Upwind Leapfrog difference scheme with limiters solves the convection problem for small Courant numbers. Thus, the proposed modification of the Upwind Leapfrog difference scheme for the numerical solution of the diffusion-convection problem has higher accuracy than other schemes, for the values of the grid Peclet number in the range $2\le Pe\le 20$, which allows you to use this class of schemes for the numerical solution of problems of computational oceanology.
Keywords: transfer problem, Upwind Leapfrog difference scheme, Standard Leapfrog difference scheme, linear-weighted combination, increase accuracy. 
@article{CHEB_2020_21_4_a20,
     author = {A. I. Sukhinov and A. E. Chistyakov and E. A. Protsenko and A. M. Atayan},
     title = {Linear combination of the {Upwind} and {Standard} {Leapfrog} difference schemes with weight coefficients, obtained by minimizing the approximation error},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {243--256},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2020_21_4_a20/}
}
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A. I. Sukhinov; A. E. Chistyakov; E. A. Protsenko; A. M. Atayan. Linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients, obtained by minimizing the approximation error. Čebyševskij sbornik, Tome 21 (2020) no. 4, pp. 243-256. http://geodesic.mathdoc.fr/item/CHEB_2020_21_4_a20/