A hybrid difference scheme under generalized approximation condition in the space of undetermined coefficients

Construction of difference schemes of high approximation orders for hyperbolic problems is still an important problem. For the construction of grid-characteristic methods, difference schemes were earlier analyzed in the space of undetermined coefficients, where the coefficients of high order derivatives in the first differential approximation of the difference scheme were used as the objective function to be minimized. Other reasonable functionals in the space of undetermined coefficients that are linear in the coefficients of the scheme may be used. By solving a linear programming problem, difference schemes meeting various conditions can be chosen. An example of the linear functional related to the approximation properties of the problem is discussed. It is proposed to call it the generalized approximation condition. Based on this condition, a difference scheme of a novel class is built that has no analogs in the literature. The presentation uses the transport equation with a constant coefficient as an example.