Monotone bicompact schemes for a linear advection equation

Monotone bicompact schemes for a linear advection equation are studied. Initial-boundary value Cauchy problem for the linear advection equation is considered and a nonuniform grid on a positively infinite interval is introduced. Assuming that any bounded monotone grid function can be represented as a linear combination of simple monotone step functions, the Godunov monotonicity for simple functions is examined. The differential problem with a discontinuous initial condition is approximated by difference scheme with the initial and boundary conditions. The Runge-Kutta scheme is found to be accurate and is A-stable and the difference is absolutely stable. The weight function is related to the behavior of the solution in such a manner that it has a value 1 near the discontinuities. The grid function at the (n + 1)th time level is calculated by integrating the function according to Simpson's rule.