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

Authors: Bojan Popov and Ognian Trifonov
Journal: Math. Comp. 75 (2006), 1735-1753
MSC (2000): Primary 65M15; Secondary 65M12
Published electronically: May 23, 2006
MathSciNet review: 2240633
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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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Additional Information

Bojan Popov
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77845

Ognian Trifonov
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Keywords: Conservation laws, error estimates, second order schemes, minmod limiter
Received by editor(s): April 22, 2004
Received by editor(s) in revised form: July 6, 2005
Published electronically: May 23, 2006
Additional Notes: The first author was supported in part by NSF DMS Grant #0510650.
The second author was supported in part by NSF DMS Grant #9970455.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.