Monotone schemes of conditional approximation and arbitrary order of accuracy for the transport equation

An initial-boundary value problem for the one-dimensional transport equation with a constant coefficient $a>0$ is approximated by a usual explicit explicit monotone difference scheme of traveling calculation “upwind scheme”. Under a Courant-type condition, it is proved that the scheme has an arbitrary $k$th order of accuracy for smooth solutions. Assuming the existence of weakly discontinuous solutions, the results are generalized to multidimensional equations. Monotone finite difference schemes for equations with variable coefficients and for first-order semilinear hyperbolic equations are constructed with the use of a special Steklov averaging with respect to nonlinearity. The efficiency of the considered methods is illustrated by numerical results.