Monotone high-precision compact scheme for quasilinear hyperbolic equations

Monotone homogeneous compact difference scheme, previously proposed by the authors for the linear transport equation, generalized to the case of quasilinear equations of hyperbolic type. The generalized scheme is fourth order approximation in spatial coordinates on a compact stencil and a first order approximation in time. The scheme is conservative, absolutely stable, monotonic over a wide range of local Courant number and can be solved by explicit formulas of the running calculation method. Quasimonotone three-stage scheme, which has the third-order approximation in time for smooth solutions, built on the basis of the scheme first-order approximation in time. Numerical results demonstrate the accuracy of the proposed schemes and their monotonicity in the solution of test problems for the quasilinear Hopf equation.