Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound

Author:
Nikolaos Sfakianakis

Journal:
Math. Comp. **82** (2013), 129-151

DOI:
https://doi.org/10.1090/S0025-5718-2012-02615-9

Published electronically:
August 16, 2012

MathSciNet review:
2983018

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Abstract | References | Additional Information

Abstract: We consider 3-point numerical schemes, that resolve scalar conservation laws, that are oscillatory either to their dispersive or anti-diffusive nature. The spatial discretization is performed over non-uniform adaptively redefined meshes. We provide a model for studying the evolution of the extremes of the oscillations. We prove that proper mesh reconstruction is able to control the oscillations; we provide bounds for the Total Variation (TV) of the numerical solution. We, moreover, prove under more strict assumptions that the increase of the TV, due to the oscillatory behavior of the numerical schemes, decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D). We finally provide numerical evidence supporting the analytical results that exhibit the stabilization properties of the mesh adaptation technique.

**[AD06]**Ch. Arvanitis and A. I. Delis,*Behavior of finite volume schemes for hyperbolic conservation laws on adaptive redistributed spatial grids*, SIAM J. Sci. Comput.**28**(2006), 1927-1956. MR**2272195 (2007m:65064)****[AKM01]**Ch. Arvanitis, Th. Katsaounis, and Ch. Makridakis,*Adaptive finite element relaxation schemes for hyperbolic conservation laws*, Math. Model. Anal. Numer.**35**(2001), 17-33. MR**1811979 (2002g:65119)****[AMS10]**Ch. Arvanitis, Ch. Makridakis, and N. Sfakianakis,*Entropy conservative schemes and adaptive mesh selection for hyperbolic conservation laws*, J. Hyp. Diff. Eq.**3**(2010), 383-404. MR**2735815 (2011k:65128)****[AMT04]**Ch. Arvanitis, Ch. Makridakis, and A. Tzavaras,*Stability and convergence of a class of finite element schemes for hyperbolic systems of conservation laws*, SIAM J. Numer. Anal.**42**(2004), 1357-1393. MR**2114282 (2005m:65204)****[Arv08]**Ch. Arvanitis,*Mesh redistribution strategies and finite element method schemes for hyperbolic conservation laws*, J. Sci. Computing**34**(2008), 1-25. MR**2367009 (2008j:65154)****[DD87]**E. Dorfi and L. Drury,*Simple adaptive grids for 1d initial value problems*, J. Computational Physics**69**(1987), 175-195.**[DZ10]**A. van Dam and P. A. Zegeling,*Balanced monitoring of flow phenomena in moving mesh methods*, Commun. Comput. Physics**7**(2010), 138-170. MR**2673131 (2011g:65157)****[For88]**B. Fornberg,*Generation of finite difference formulas on arbitrary spaced grids*, Mathematics of Computations**51**(1988), 699-706. MR**935077 (89b:65055)****[HH83]**A. Harten and J. Hyman,*Self adjusting grid methods for one-dimensional hyperbolic conservation laws*, J. Comput. Physics**50**(1983), 235-269. MR**707200 (85g:65111)****[HR10]**W. Huang and R.D. Russel,*Adaptive moving mesh methods*, Springer, 2010. MR**2722625****[Kro97]**D. Kroener,*Numerical schemes for conservation laws*, Wiley Teubner, 1997. MR**1437144 (98b:65003)****[LeV92]**R. LeVeque,*Numerical methods for conservation laws*, second ed., Birkhäuser Verlag, 1992. MR**1153252 (92m:65106)****[LeV02]**-,*Finite volume methods for hyperbolic problems*, first ed., Cambridge Texts in Applied Mathematics, 2002. MR**1925043 (2003h:65001)****[LR56]**P. D. Lax and R. Richtmyer,*Survey of the stability of linear finite difference equations*, Comm. Pure Appl. Math.**9**(1956), 267-293. MR**0079204 (18:48c)****[Luc85]**B. J. Lucier,*A stable adaptive numerical scheme for hyperbolic conservation laws*, SIAM J. Num. Anal.**22**(1985), 180-203. MR**772891 (86d:65123)****[Luc86]**-,*A moving mesh numerical method for hyperbolic conservation laws*, Math. Comput.**46**(1986), no. 173, 59-69. MR**815831 (87m:65141)****[LW60]**P. D. Lax and B. Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217-237. MR**0120774 (22:11523)****[Sfa09]**N. Sfakianakis,*Finite difference schemes on non-uniform meshes for hyperbolic conservation laws*, Ph.D. thesis, University of Crete, 2009.**[TT03]**H. Tang and T. Tang,*Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws*, SIAM J. Numerical Analysis**41**(2003), 487-515. MR**2004185 (2004f:65143)**

Additional Information

**Nikolaos Sfakianakis**

Affiliation:
University of Vienna, Austria

Address at time of publication:
Johannes Gutenberg University, Mainz, Germany

Email:
sfakiana@uni-mainz.de

DOI:
https://doi.org/10.1090/S0025-5718-2012-02615-9

Received by editor(s):
September 19, 2009

Received by editor(s) in revised form:
September 2, 2011

Published electronically:
August 16, 2012

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.