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

Popov, Bojan; Trifonov, Ognian

doi:10.1090/S0025-5718-06-01875-8

Order of convergence of second order schemes based on the minmod limiter
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by Bojan Popov and Ognian Trifonov PDF

Math. Comp. 75 (2006), 1735-1753 Request permission

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 $L_p$-error of monotone finite difference methods for the linear advection equation is of order $1/2$ for initial data in $W^1(L_p)$, $1\leq p\leq \infty$. 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 $L_2$-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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