On a family of monotone finite-difference schemes of the second order of approximation

Using a simple model of a linear transport equation a family of hybrid monotone finite difference schemes has been constructed. By the analysis of the differential approximation it was shown that the resulting family has a secondorder approximation in the spatial variable, has minimal scheme viscosity and dispersion and monotonous. It is shown that the region of operability of the base schemes (Modified Central Difference Schemes (MCDS) and Modified Upwind Difference Schemes (MUDS)) is a non-empty set. The local criterion for switching between the base schemes is based on the sign of the product of the velocity, the first and second differences of the transferred functions at the considered point. On the solution of the Cauchy problem provides a graphical comparison of the calculation results obtained using the known schemes of the first, second and third order approximation. This work has been partly supported by Russian Foundation for Basic Research (grants No. 14-01-00428, 15-51-50023), by the program of the Presidium of RAS No. 8 and by the program No. 3 of the Department of Mathematical Sciences of RAS.