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Computation of fluxes of conservation laws. (English) Zbl 1193.65155
Author’s abstract: The direct method for the construction of local conservation laws of partial differential equations (PDEs) is a systematic method applicable to a wide class of PDE systems [S. Anco and G. Bluman, Eur. J. Appl. Math. 13, No. 5, 567–585 (2002; Zbl 1034.35071)]. 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.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35B06 Symmetries, invariants, etc. in context of PDEs
68W30 Symbolic computation and algebraic computation
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
Full Text: DOI arXiv
[1] Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Commun Pure Appl Math 21: 467–490 · Zbl 0162.41103
[2] Benjamin TB (1972) The stability of solitary waves. Proc R Soc Lond A 328: 153–183
[3] Knops RJ, Stuart CA (1984) Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch Ration Mech Anal 86: 234–249 · Zbl 0589.73017
[4] LeVeque RJ (1992) Numerical methods for conservation laws. Birkhäuser, Basel · Zbl 0847.65053
[5] Godlewski E, Raviart P-A (1996) Numerical approximation of hyperbolic systems of conservation laws. Springer, Berlin · Zbl 0860.65075
[6] Bluman G, Kumei S (1987) On invariance properties of the wave equation. J Math Phys 28: 307–318 · Zbl 0662.35065
[7] Bluman G, Kumei S, Reid G (1988) New classes of symmetries for partial differential equations. J Math Phys 29: 806–811 · Zbl 0669.58037
[8] Bluman G, Cheviakov AF (2005) Framework for potential systems and nonlocal symmetries: algorithmic approach. J Math Phys 46: 123506 · Zbl 1111.35002
[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 · Zbl 1112.35010
[10] Sjöberg A, Mahomed FM (2004) Non-local symmetries and conservation laws for one-dimensional gas dynamics equations. Appl Math Comput 150: 379–397 · Zbl 1102.76059
[11] Akhatov S, Gazizov R, Ibragimov N (1991) Nonlocal symmetries. Heuristic approach. J Sov Math 55: 1401–1450 · Zbl 0760.35002
[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 · Zbl 1157.35002
[13] Noether E (1918) Invariante Variationsprobleme. Nachr König Gesell Wissen Göttingen, Math-Phys Kl 235–257 · JFM 46.0770.01
[14] Bluman G (2005) Connections between symmetries and conservation laws. Symm Integr Geom: Meth Appl (SIGMA) 1:011, 16 pages · Zbl 1092.70016
[15] Wolf T (2002) A comparison of four approaches to the calculation of conservation laws. Eur J Appl Math 13(2): 129–152 · Zbl 1002.35008
[16] Anco S, Bluman G (1997) Direct construction of conservation laws. Phys Rev Lett 78: 2869–2873 · Zbl 0948.58015
[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 · Zbl 1034.35071
[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) · Zbl 1253.35007
[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 · Zbl 1161.65376
[20] Anco S (2003) Conservation laws of scaling-invariant field equations. J Phys A: Math Gen 36: 8623–8638 · Zbl 1063.70024
[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/\(\sim\)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) · Zbl 1182.65044
[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/\(\sim\)bernard/papers.html · Zbl 1182.65044
[27] Olver PJ (1983) Conservation laws and null divergences. Math Proc Camb Phil Soc 94: 529–540 · Zbl 0556.35021
[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 · Zbl 1114.80306
[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) · Zbl 1189.76759
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