Computation of fluxes of conservation laws
The direct method for the construction of local conservation laws of partial differential equations (PDE) is a systematic method applicable to a wide class of PDE systems (S. Anco and G. Bluman, Eur J Appl Math 13:567–585, 2002). According to the direct method one seeks multipliers, such that the linear combination of PDEs of a given system with these multipliers yields a divergence expression. Once local-conservation-law multipliers have been found, one needs to reconstruct the fluxes of the conservation law. In this review paper, common methods of flux computation are discussed, compared, and illustrated by examples. An implementation of these methods in symbolic software is also presented.
KeywordsConservation laws Direct construction method Multipliers Symbolic software
Unable to display preview. Download preview PDF.
- 13.Noether E (1918) Invariante Variationsprobleme. Nachr König Gesell Wissen Göttingen, Math-Phys Kl 235–257Google Scholar
- 14.Bluman G (2005) Connections between symmetries and conservation laws. Symm Integr Geom: Meth Appl (SIGMA) 1:011, 16 pagesGoogle Scholar
- 18.Bluman G, Cheviakov AF, Anco S (2009) Construction of conservation laws: how the direct method generalizes Noether’s theorem. In: Proceedings of 4th workshop group analysis of differential equations & integrability (to appear)Google Scholar
- 19.Hereman W, Colagrosso M, Sayers R, Ringler A, Deconinck B, Nivala M, Hickman MS (2005) Continuous and discrete homotopy operators and the computation of conservation laws. In: Wang D, Zheng Z (eds) Differential equations with symbolic computation. Birkhäuser Verlag, Boston, pp 249–285Google Scholar
- 21.Bluman GW, Cheviakov AF, Anco SC (2009) Advanced symmetry methods for partial differential equations. Appl Math Sci ser (to appear)Google Scholar
- 22.Cheviakov AF (2007) GeM software package for computation of symmetries and conservation laws of differential equations. Comput Phys Commun 176(1):48–61. (In the current paper, we used a new version of GeM software, which is scheduled for public release in 2009. See http://math.usask.ca/~cheviakov/gem/)Google Scholar
- 23.Wolf T (2002) Crack, LiePDE, ApplySym and ConLaw, section 4.3.5 and computer program on CD-ROM. In: Grabmeier J, Kaltofen E, Weispfenning V (eds) Computer algebra handbook. Springer, Berlin, pp 465–468Google Scholar
- 24.Hereman W, TransPDEDensityFlux.m, PDEMultiDimDensityFlux.m, and DDEDensityFlux.m: Mathematica packages for the symbolic computation of conservation laws of partial differential equations and differential-difference equations. Available from the software section at http://www.mines.edu/fs_home/whereman/
- 25.Deconinck B, Nivala M (2008) Symbolic integration using homotopy methods. Preprint, Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420. Math Comput Simul (in press)Google Scholar
- 26.Deconinck B, Nivala M Maple software for the symbolic computation of conservation laws of (1 + 1)-dimensional partial differential equations. http://www.amath.washington.edu/~bernard/papers.html
- 29.Oberlack M, Cheviakov AF (2009) Higher-order symmetries and conservation laws of the G-equation for premixed combustion and resulting numerical schemes. J Eng Math (Submitted)Google Scholar