Family of quasi-monotonic finite-difference schemes of the second-order of approximation

Using a simple model of a linear transfer equation, a family of hybrid monotonic finite-difference schemes is constructed. By differential approximation analysis, it is shown that the resulting family yields a second-order approximation in the spatial variable, having minimal scheme viscosity and dispersion and being monotonic. It is demonstrated that the operability domain of the basic schemes, namely, the modified central difference schemes (MCDS) and the modified upwind difference schemes (MUDS), forms a nonempty set. A local criterion for switching between the basic schemes is proposed; this criterion employs the sign of the product of the velocity, as well as the first and second differences of the transferred functions at the considered point. Within the studied schemes, the optimal pair of basic schemes, possessing the above-mentioned properties and being closest to the third-order scheme, is obtained. On the solution of the Cauchy problem, the calculation results obtained using some well-known first-, second-, and third-order schemes are compared graphically.