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Modified equation for adaptive monotone difference schemes and its convergent analysis


Author: Zhen-Huan Teng
Journal: Math. Comp. 77 (2008), 1453-1465
MSC (2000): Primary 65M06, 65M15, 35K15
DOI: https://doi.org/10.1090/S0025-5718-08-02061-9
Published electronically: January 24, 2008
MathSciNet review: 2398776
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Abstract: A modified parabolic equation for adaptive monotone difference schemes based on equal-arclength mesh, applied to the linear convection equation, is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous (piecewise smooth) solution of the linear convection equation at order one rate in the $ L_1$-norm. It is well known that solutions of the monotone schemes with uniform meshes and their modified equation approach the same discontinuous solution at a half-order rate in the $ L_1$-norm. Therefore, the convergence analysis for the modified equation provided in this work demonstrates theoretically that the monotone schemes with adaptive grids can improve the solution accuracy. Numerical experiments also confirm the theoretical conclusions.


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Additional Information

Zhen-Huan Teng
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
Email: tengzh@math.pku.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-08-02061-9
Keywords: Error estimates, adaptive monotone schemes, modified parabolic equation, convection equation.
Received by editor(s): August 2, 2006
Received by editor(s) in revised form: April 17, 2007
Published electronically: January 24, 2008
Additional Notes: This work was supported in part by the National Natural Science Foundation of China (10576001).
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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