Order of convergence of second order schemes based on the minmod limiter

Many second order accurate nonoscillatory schemes are based on the minmod limiter, e.g., the Nessyahu-Tadmor scheme. It is well known that the -error of monotone finite difference methods for the linear advection equation is of order for initial data in , . For second or higher order nonoscillatory schemes very little is known because they are nonlinear even for the simple advection equation. In this paper, in the case of a linear advection equation with monotone initial data, it is shown that the order of the -error for a class of second order schemes based on the minmod limiter is of order at least in contrast to the order for any formally first order scheme.