×

A linear-algebraic method to compute polynomial PDE conservation laws. (English) Zbl 1475.35124

Summary: We present a method to compute polynomial conservation laws for systems of partial differential equations (PDEs). The method only relies on linear algebraic computations and is complete, in the sense it can find a basis for all polynomial fluxes that yield conservation laws, up to a specified order of derivatives and degree. We compare our method to state-of-the-art algorithms based on the direct approach on a few PDE systems drawn from mathematical physics.

MSC:

35G50 Systems of nonlinear higher-order PDEs
35L65 Hyperbolic conservation laws
68W30 Symbolic computation and algebraic computation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[2] Anco, S.; Bluman, G., Direct construction method for conservation laws of partial differential equations. Part II: general treatment, Eur. J. Appl. Math., 13, 567-585 (2002) · Zbl 1034.35071
[3] Boreale, M., Complete algorithms for algebraic strongest postconditions and weakest preconditions in polynomial odes, Elsevier. Elsevier, Sci. Comput. Program.. (SOFSEM 2018. SOFSEM 2018, LNCS, vol. 10706 (2018), Springer), 193, 442-455 (2020), Short version in · Zbl 1450.12004
[4] Boreale, M., On the coalgebra of partial differential equations, (MFCS 2019. MFCS 2019, LIPIcs, vol. 138 (2019), Schloss Dagstuhl-Leibniz-Zentrum für Informatik), 24:1-24:13 · Zbl 07561668
[5] Boulier, F.; Lemaire, F., Finding first integrals using normal forms modulo differential regular chains, (Computer Algebra in Scientific Computing, CASC 2015. Computer Algebra in Scientific Computing, CASC 2015, LNCS, vol. 9301 (2015), Springer: Springer Aachen, Germany), 101-118 · Zbl 1434.12003
[6] Cheviakov, A., Computation of fluxes of conservation laws, J. Eng. Math., 66, 153-173 (2010) · Zbl 1193.65155
[7] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms an Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics (2007), Springer · Zbl 1118.13001
[8] Hereman, W.; Adams, P. J.; Eklund, H. L.; Hickman, M. S.; Herbst, B. M., Direct methods and symbolic software for conservation laws of nonlinear equations, (Yan, Z., Advances in Nonlinear Waves and Symbolic Computation (2009), Nova Science Publishers: Nova Science Publishers New York), 19-79 · Zbl 1210.35164
[9] Hereman, W.; Colagrosso, M.; Sayers, R.; Ringler, A.; Deconinck, B.; Nivala, M.; Hickman, M. S., Continuous and discrete homotopy operators and the computation of conservation laws, (Wang, D.; Zheng, Z., Differential Equations with Symbolic Computation (2005), Birkhäuser: Birkhäuser Basel), 249-285
[10] Lemaire, F., Les classements les plus généraux assurant l’analycité des solutions des systèmes orthonomes pour des conditions initiales analytiques, (Ganzha, V. G.; Mayr, E. W.; Vorozhtsov, E. V., Computer Algebra in Scientific Computing, CASC 2002. Proceedings of the Fifth International Workshop on Computer Algebra in Scientific Computing, Yalta, Ukraine (2002), Technische Universität München: Technische Universität München Germany)
[11] LeVeque, R. J., Numerical Methods for Conservation Laws (1992), Birkhäuser · Zbl 0847.65053
[12] Naz, R.; Mahomed, F. M.; Mason, D. P., Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Appl. Math. Comput., 205, 212-230 (2008) · Zbl 1153.76051
[13] Olver, P. J., Applications of Lie Groups to Differential Equations, 2/E. Graduate Texts in Mathemathics (1993), Springer · Zbl 0785.58003
[14] Poole, D.; Hereman, W., Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions, J. Symb. Comput., 46, 12, 1355-1377 (2011) · Zbl 1234.35071
[15] Riquier, C., Les systèmes d’équations aux dérivèes partielles (1910), Gauthiers-Villars: Gauthiers-Villars Paris · JFM 40.0411.01
[16] Rust, C. J.; Reid, G. J.; Wittkopf, A. D., Existence and uniqueness theorems for formal power series solutions of analytic differential systems, (ISSAC 1999 (1999), ACM), 105-112
[17] Wolf, T., A comparison of four approaches to the calculation of conservation laws, Eur. J. Appl. Math., 13, 129-152 (2002) · Zbl 1002.35008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.