Method for constructing high-order approximation schemes for hyperbolic equations

A method for constructing high-order difference schemes of approximation is proposed for solving the simplest hyperbolic type equation, namely, for the linear transport equation. Based on the developed method, the Rusanov, Warming-Cutler-Lomax schemes are analyzed and new third-order difference schemes are constructed. For the difference schemes proposed in this paper, a method for monotonizing the solution is proposed. The monotonization of the numerical solution is carried out by lowering the order of the difference scheme at the points of oscillation of the numerical solution. This is achieved by nesting the template of lower spatial derivatives, which are a subset of the templates of difference schemes of higher derivatives according to the “matryoshka principle”. The results of numerical experiments for known test problems are presented.