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

Many second order accurate non-oscillatory schemes are based on the Minmod limiter, for example the Nessyahu-Tadmor scheme. It is well known that the Lperror of monotone finite difference methods for the linear advection equation is of order 1/2 for initial data in W (Lp), 1 ≤ p ≤ ∞, see [2]. For a second or higher order non-oscillatory 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 L2-error for a class of second order schemes based on the Minmod limiter is of order at least 5/8 in contrast to the 1/2 order for any formally first order scheme. AMS subject classification: Primary 65M15; Secondary 65M12