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Error bound between monotone difference schemes and their modified equations

Author: Zhen-Huan Teng
Journal: Math. Comp. 79 (2010), 1473-1491
MSC (2000): Primary 65M06, 65M15; Secondary 35L45, 35K15
Published electronically: September 14, 2009
MathSciNet review: 2630000
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Abstract: It is widely believed that if monotone difference schemes are applied to the linear convection equation with discontinuous initial data, then solutions of the monotone schemes are closer to solutions of their parabolic modified equations than that of the original convection equation. We will confirm the conjecture in this paper. It is well known that solutions of the monotone schemes and their parabolic modified equations approach discontinuous solutions of the linear convection equation at a rate only half in the $ L^1$-norm. We will prove that the error bound between solutions of the monotone schemes and that of their modified equations is order one in the $ L^1$-norm. Therefore the conclusion shows that the monotone schemes solve the modified equations more accurately than the original convection equation even if the initial data is discontinuous. As a consequence of the main result, we will show that the half-order rate of convergence for the monotone schemes to the convection equation is the best possible.

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Additional Information

Zhen-Huan Teng
Affiliation: LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, China

Keywords: Monotone difference schemes, modified parabolic equation, discontinuous solutions, $L^1$-error estimates, linear convection equation
Received by editor(s): February 2, 2009
Received by editor(s) in revised form: May 20, 2009
Published electronically: September 14, 2009
Additional Notes: This work was supported by the National Science Foundation of China (No. 10576001)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.