Monotone compact running schemes for systems of hyperbolic equations

For quasilinear hyperbolic equations, conservative absolutely stable compact schemes are presented that are monotone in a wide range of local Courant numbers. The schemes are fourth-order accurate in space on a compact stencil and first-or third-order accurate in time. They are efficient and are solved by the running calculation method. The convergence rate of the schemes is analyzed in detail in the case of mesh refinement for solutions of various orders of smoothness. The capabilities of the schemes are demonstrated by solving well-known one-dimensional test problems for gas dynamics equations.