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

Abstract

Many second order accurate nonoscillatory schemes are based on the minmod limiter, e.g., the Nessyahu–Tadmor scheme. It is well known that the L p L_p -error of monotone finite difference methods for the linear advection equation is of order 1 / 2 1/2 for initial data in W 1 ( L p ) W^1(L_p) , 1 ≤ p ≤ ∞ 1\leq p\leq \infty . 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 L_2 -error for a class of second order schemes based on the minmod limiter is of order at least 5 / 8 5/8 in contrast to the 1 / 2 1/2 order for any formally first order scheme.