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 Lp-error of monotone finite difference methods for the linear advection equation is of order 1/2 for initial data in W 1 (Lp), 1 ≤ p < 00. 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 L 2 -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.

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