Conservation laws and symmetrization of the equations of incompressible inviscid fluids
Author:
Lubin G. Vulkov
Journal:
Quart. Appl. Math. 57 (1999), 549-560
MSC:
Primary 35L65; Secondary 76B99
DOI:
https://doi.org/10.1090/qam/1704427
MathSciNet review:
MR1704427
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Abstract: Many physical processes are described by conservative quasilinear systems of equations that admit a supplementary independent conservation law. A typical example is the equations of incompressible fluid dynamics. If for basic equations the conservation laws of the momentum and the volume are chosen then the conservation law of the energy is an additional conservation law. In the present paper all conservation laws for incompressible fluid dynamics are derived and the symmetric forms of its equations are obtained. Godunov’s method of symmetrization is generalized for the case of an arbitrary quasilinear system of equations.
L. I. Judovič, Nonstationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz. 3, No. 6, 1032–1066 (1963); English transl. in USSR Comput. Math. and Math. Phys. 3 (1963)
- Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis 20 (1975), no. 1, 32–43. MR 0430568, DOI https://doi.org/10.1016/0022-1236%2875%2990052-x
S. N. Antoncev, A. V. Kažihov, and V. M. Monahov, Boundary Value Problems of Nonhomogeneous Fluid Mechanics, Nauka, Novosibirsk, 1983; English transl., North-Holland, Amsterdam, 1990
R. Peyret and T. O. Taylor, Computational Methods in Fluid Flow, Springer, New York, 1982
- Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734
- L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Y. Chapovsky; Translation edited by William F. Ames. MR 668703
- Nail H. Ibragimov, Transformation groups applied to mathematical physics, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1985. Translated from the Russian. MR 785566
- P. R. Eiseman and A. P. Stone, Conservation laws of fluid dynamics—a survey, SIAM Rev. 22 (1980), no. 1, 12–27. MR 554708, DOI https://doi.org/10.1137/1022002
- B. L. Roždestvenskiĭ and N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, Translations of Mathematical Monographs, vol. 55, American Mathematical Society, Providence, RI, 1983. Translated from the second Russian edition by J. R. Schulenberger. MR 694243
- D. Serre, Les invariants du premier ordre de l’equation d’Euler en dimension trois, Phys. D 13 (1984), no. 1-2, 105–136 (French, with English summary). MR 775281, DOI https://doi.org/10.1016/0167-2789%2884%2990273-2
- E. D. Terent′ev and Ju. D. Šmyglevskiĭ, A complete system of equations in divergence form for the electromagnetic dynamics of a perfect gas, Ž. Vyčisl. Mat i Mat. Fiz. 16 (1976), no. 3, 725–737, 821 (Russian). MR 452108
- Lubin G. Vulkov, On the conservation laws in magnetohydrodynamics, Appl. Anal. 75 (2000), no. 1-2, 1–18. MR 1784808, DOI https://doi.org/10.1080/00036810008840831
- L. G. Volkov, Conservation laws of gas dynamics in Lagrangian variables, Zh. Vychisl. Mat. i Mat. Fiz. 31 (1991), no. 10, 1552–1562 (Russian); English transl., Comput. Math. Math. Phys. 31 (1991), no. 10, 93–101 (1992). MR 1145224
- L. G. Vulkov, Conservation laws in nonlinear elasticity. I. One-dimensional elastodynamics, Quart. Appl. Math. 52 (1994), no. 3, 427–438. MR 1292195, DOI https://doi.org/10.1090/qam/1292195
- Lubin G. Vulkov, On the conservation laws of the compressible Euler equations, Appl. Anal. 64 (1997), no. 3-4, 255–271. MR 1460082, DOI https://doi.org/10.1080/00036819708840534
- S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). MR 0131653
- Amiram Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys. 49 (1983), no. 1, 151–164. MR 694161, DOI https://doi.org/10.1016/0021-9991%2883%2990118-3
S. K. Godunov, Symmetric form of the magnetohydrodynamic equations, Čisl. Metody Meh. Šplošnoi Sredy 3, No. 1, 26–34 (1972) (in Russian)
- A. A. Samarskiĭ and Yu. P. Popov, Raznostnye metody resheniya zadach gazovoĭ dinamiki, 2nd ed., “Nauka”, Moscow, 1980 (Russian). MR 619540
Yu. I. Shokin, On conservatism of difference schemes of gas dynamics, Lecture Notes in Physics, vol. 264, 1986, pp. 578–583
- L. G. Volkov, On the construction of completely conservative difference schemes for equations in the nonlinear theory of elasticity, Zh. Vychisl. Mat. i Mat. Fiz. 31 (1991), no. 9, 1392–1401 (Russian); English transl., Comput. Math. Math. Phys. 31 (1991), no. 9, 92–99 (1992). MR 1145208
L. I. Judovič, Nonstationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz. 3, No. 6, 1032–1066 (1963); English transl. in USSR Comput. Math. and Math. Phys. 3 (1963)
R. Temam, On the Euler equations of an incompressible perfect fluid, J. Functional Analysis 20, No. 1, 32–43 (1975)
S. N. Antoncev, A. V. Kažihov, and V. M. Monahov, Boundary Value Problems of Nonhomogeneous Fluid Mechanics, Nauka, Novosibirsk, 1983; English transl., North-Holland, Amsterdam, 1990
R. Peyret and T. O. Taylor, Computational Methods in Fluid Flow, Springer, New York, 1982
P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982
N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985
P. R. Eiseman and A. P. Stone, Conservation laws of fluid dynamics, a survey, SIAM Review 22 12–27 (1980)
B. L. Roždestvenskii and N. N. Janenko, Systems of Quasilinear Equations and their Applications to Gas Dynamics, Nauka, Moscow, 1968; English transl. of 2nd ed., Amer. Math. Soc., Providence, RI, 1983
D. Serre, Les invariants du premier ordre de l’équation d’Euler en dimension trois, Phys. D 13, 105–136 (1984)
D. Terentev and J. D. Šmyglevskii, A complete set of equations in divergence form for the electromagnetic dynamics of a perfect gas, Ž. Vyčisl. Mat. i Mat. Fiz. 25, No. 3, 725–737 (1976); English transl. in USSR Comput. Math. and Math. Phys. 16 (1976)
L. G. Vulkov,$^{1}$ On the conservation laws in magnetohydrodynamics at infinite conductivity, Ž. Vyčisl. 25, No. 7, 1401–1408 (1985); English transl. in USSR Comput. Math. and Math. Phys. 25 (1985)
L. G. Vulkov,$^{1}$ Conservation laws of the two-dimensional gas dynamics equations in Lagrange variables, Ž. Vyčisl. Mat. i Mat. Fiz. 3, No. 10, 1552–1562 (1991); English transl. in USSR Comput. Math. and Math. Phys. 31 (1991)
L. G. Vulkov, Conservation laws in nonlinear elasticity, I. One-dimensional elastodynamics, Quart. Appl. Math. 52 427–438 (1994)
L. G. Vulkov, On the conservation laws of the compressible Euler equations, Applicable Analysis 64, No. 3–4, 255–271 (1997)
S. K. Godunov, An interesting class of quasilinear systems, DAN SSSR 139, No. 3, 521–523 (1961); English transl. in Soviet Math. Dokl. 2 (1961)
A. Harten, On the symmetric form of systems of conservation laws with entropy, J. Comp. Phys. 49, 151–169 (1989)
S. K. Godunov, Symmetric form of the magnetohydrodynamic equations, Čisl. Metody Meh. Šplošnoi Sredy 3, No. 1, 26–34 (1972) (in Russian)
A. A. Samarskii and Yu. P. Popov, Difference methods for the solution of problems of gas dynamics, M: Nauka, 1980 (in Russian)
Yu. I. Shokin, On conservatism of difference schemes of gas dynamics, Lecture Notes in Physics, vol. 264, 1986, pp. 578–583
L. G. Vulkov$^{1}$, On construction of fully conservative difference schemes of the nonlinear elasticity equations, Ž. Vyčisl. Mat. i Mat. Fiz. 31, No. 9, 1392–1401 (1991); English transl. in USSR Comput. Math. and Math. Phys. 31 (1991)
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