Construction of monotone difference schemes for systems of hyperbolic equations

A distinctive feature of hyperbolic equations is the finite propagation velocity of perturbations in the region of integration (wave processes) and the existence of characteristic manifolds: characteristic lines and surfaces (bounding the domains of dependence and influence of solutions). Another characteristic feature of equations and systems of hyperbolic equations is the appearance of discontinuous solutions in the nonlinear case even in the case of smooth (including analytic) boundary conditions: the so-called gradient catastrophe. In this paper, on the basis of the characteristic criterion for monotonicity, a universal algorithm is proposed for constructing high-order schemes monotone for arbitrary form of the sought-for solution, based on their analysis in the space of indefinite coefficients. The constructed high-order difference schemes are tested on the basis of the characteristic monotonicity criterion for nonlinear one-dimensional systems of hyperbolic equations.