Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Modified equation for adaptive monotone difference schemes and its convergent analysis

Author(s): Zhen-Huan Teng.
Journal: Math. Comp. 77 (2008), 1453-1465.
MSC (2000): Primary 65M06, 65M15, 35K15
Posted: January 24, 2008
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

1.
F. Bouchut and B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), 2847-2870. MR 1475677 (98m:65156)

2.
B. Despres, Discrete compressive solutions of scalar conservation laws, Journal of Hyperbolic Differential Equations 3 (2004), 493-520. MR 2094527 (2006a:35194)

3.
A. Harten, The artificial compression method for computation of shocks and contact discontinuities: III. Self-adjusting hybrid schemes, Math. Comp. 32 (1978), 363-389. MR 0489360 (58:8789)

4.
A. Harten, J. M. Hyman and P. D. Lax, On finite-difference approximation and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297-322. MR 0413526 (54:1640)

5.
W. Huang, Y. Ren and R. D. Russell, Moving mesh partial differential equations (MMPDES) based on the equidistribution principle, SIAM J. Numer. Anal. 31 (1994), 709-730. MR 1275109 (94m:65149)

6.
N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comp. Math. and Math. Phys. 16 (1976), 105-119.

7.
O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23 (1968).

8.
G. M. Leiberman, Second Order Parabolic Differential Equations, World Scientific, (1998).

9.
K. Lipnikov and M. Shashkov, The error-minimization-based strategy for moving mesh methods. Commun. Comput. Phys., 1 (2006), 53-80.

10.
J. M. Stockie, J. A. Mackkenzie and R. D. Russel, A moving mesh method for one-dimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 22 (2001), 1791-1813. MR 1813298 (2001m:65116)

11.
H. Z. Tang and T. Tang, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003), 487-515.

MR 2004185 (2004f:65143)

12.
T. Tang and Z. H. Teng, The sharpness of Kuznetsov's $ O(\sqrt{\Delta x}\,)$ $ L^1$-error estimate for monotone difference schemes, Math. Comp. 64 (1995), 581-589. MR 1270625 (95f:65176)

13.
T. Tang and Z. H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp. 66 (1997), 495-526. MR 1397446 (97m:65160)

14.
Z. H. Teng and P. W. Zhang, Optimal $ L^1$-rate of convergence for viscosity method and monotone scheme to piecewise constant solutions with shocks, SIAM J. Numer. Anal. 34 (1997), 959-978. MR 1451109 (98f:65094)

15.
Z. Zhang, Moving mesh method with conservative interpolation based on L2-projection, Commun. Comput. Phys. 1 (2006), 930-944.

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 65M06, 65M15, 35K15

Retrieve articles in all Journals with MSC (2000): 65M06, 65M15, 35K15

Additional Information:

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

DOI: 10.1090/S0025-5718-08-02061-9
PII: S 0025-5718(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
Posted: January 24, 2008
Additional Notes: This work was supported in part by the National Natural Science Foundation of China (10576001).
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google