Computation of fluxes of conservation laws Authors Alexei F. Cheviakov Department of Mathematics and Statistics University of Saskatchewan Article

First Online: 09 July 2009 Received: 31 January 2009 Accepted: 17 June 2009 DOI :
10.1007/s10665-009-9307-x

Cite this article as: Cheviakov, A.F. J Eng Math (2010) 66: 153. doi:10.1007/s10665-009-9307-x
Abstract 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.

Keywords Conservation laws Direct construction method Multipliers Symbolic software Download to read the full article text

References 1.

Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Commun Pure Appl Math 21: 467–490

MATH CrossRef MathSciNet 2.

Benjamin TB (1972) The stability of solitary waves. Proc R Soc Lond A 328: 153–183

CrossRef MathSciNet ADS 3.

Knops RJ, Stuart CA (1984) Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch Ration Mech Anal 86: 234–249

CrossRef MathSciNet 4.

LeVeque RJ (1992) Numerical methods for conservation laws. Birkhäuser, Basel

MATH 5.

Godlewski E, Raviart P-A (1996) Numerical approximation of hyperbolic systems of conservation laws. Springer, Berlin

MATH 6.

Bluman G, Kumei S (1987) On invariance properties of the wave equation. J Math Phys 28: 307–318

MATH CrossRef MathSciNet ADS 7.

Bluman G, Kumei S, Reid G (1988) New classes of symmetries for partial differential equations. J Math Phys 29: 806–811

MATH CrossRef MathSciNet ADS 8.

Bluman G, Cheviakov AF (2005) Framework for potential systems and nonlocal symmetries: algorithmic approach. J Math Phys 46: 123506

CrossRef MathSciNet ADS 9.

Bluman G, Cheviakov AF, Ivanova NM (2006) Framework for nonlocally related PDE systems and nonlocal symmetries: extension simplification and examples. J Math Phys 47: 113505

CrossRef MathSciNet ADS 10.

Sjöberg A, Mahomed FM (2004) Non-local symmetries and conservation laws for one-dimensional gas dynamics equations. Appl Math Comput 150: 379–397

MATH CrossRef MathSciNet 11.

Akhatov S, Gazizov R, Ibragimov N (1991) Nonlocal symmetries. Heuristic approach. J Sov Math 55: 1401–1450

CrossRef 12.

Anco SC, Bluman GW, Wolf T (2008) Invertible mappings of nonlinear PDEs to linear PDEs through admitted conservation laws. Acta Appl Math 101: 21–38

MATH CrossRef MathSciNet 13.

Noether E (1918) Invariante Variationsprobleme. Nachr König Gesell Wissen Göttingen, Math-Phys Kl 235–257

14.

Bluman G (2005) Connections between symmetries and conservation laws. Symm Integr Geom: Meth Appl (SIGMA) 1:011, 16 pages

15.

Wolf T (2002) A comparison of four approaches to the calculation of conservation laws. Eur J Appl Math 13(2): 129–152

MATH CrossRef 16.

Anco S, Bluman G (1997) Direct construction of conservation laws. Phys Rev Lett 78: 2869–2873

MATH CrossRef MathSciNet ADS 17.

Anco S, Bluman G (2002) Direct construction method for conservation laws of partial differential equations Part II: general treatment. Eur J Appl Math 13: 567–585

MATH MathSciNet 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)

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 249–285

20.

Anco S (2003) Conservation laws of scaling-invariant field equations. J Phys A: Math Gen 36: 8623–8638

MATH CrossRef MathSciNet ADS 21.

Bluman GW, Cheviakov AF, Anco SC (2009) Advanced symmetry methods for partial differential equations. Appl Math Sci ser (to appear)

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/ )

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 465–468

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)

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
27.

Olver PJ (1983) Conservation laws and null divergences. Math Proc Camb Phil Soc 94: 529–540

MATH CrossRef MathSciNet 28.

Oberlack M, Wenzel H, Peters N (2001) On symmetries and averaging of the G-equation for premixed combustion. Combust Theor Model 5: 363–383

MATH CrossRef MathSciNet ADS 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)

© Springer Science+Business Media B.V. 2009