Monotonicity criteria for difference schemes designed for hyperbolic equations

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S.K. Godunov’s, A. Harten’s (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations.