Modified equation for adaptive monotone difference schemes and its convergent analysis
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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.References
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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
- 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).
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 1453-1465
- MSC (2000): Primary 65M06, 65M15, 35K15
- DOI: https://doi.org/10.1090/S0025-5718-08-02061-9
- MathSciNet review: 2398776