Monotone high-accuracy compact running scheme for quasi-linear hyperbolic equations

Abstract

The monotone homogeneous bicompact difference scheme earlier proposed by the authors for the linear transport equation is generalized to the case of quasi-linear hyperbolic equations. The generalized scheme is of the fourth order of approximation in the spatial coordinates on a compact stencil and has the first order of approximation in time. The scheme is conservative, absolutely stable, monotone over a wide range of local Courant number values and can be solved by explicit formulas of the running calculation. A quasi-monotone three-stage scheme, which has the third-order approximation in time for smooth solutions, is constructed on the basis of the scheme with a first-order time approximation. Numerical results are presented demonstrating the accuracy of the proposed schemes and their monotonicity in the solution of test problems for the quasi-linear Hopf equation.