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A new finite element formulation for computational fluid dynamics. I: Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. (English) Zbl 0572.76068

Results of Harten and Tadmor [e.g.: A. Harten, J. Comput. Phys. 49, 151-164 (1983; Zbl 0503.76088)] are generalized to the compressible Navier-Stokes equations including heat conducting effects. A symmetric form of the equations is derived in terms of entropy variables. It is shown that finite element methods based upon this form automatically satisfy the second law of thermodynamics and that stability of the discrete solution is thereby guaranteed ab initio.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
80A20 Heat and mass transfer, heat flow (MSC2010)

Citations:

Zbl 0503.76088
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References:

[1] Harten, A., On the symmetric form of systems of conservation laws with entropy, J. comput. phys., 49, 151-164, (1983) · Zbl 0503.76088
[2] Hughes, T.J.R.; Marsden, J.E., A short course in fluid mechanics, (1976), Publish or Perish Boston, MA · Zbl 0329.76001
[3] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0545.73031
[4] Tadmor, E., Skew-selfadjoint form for systems of conservation laws, J. math. anal. appl., 103, 428-442, (1984) · Zbl 0599.35102
[5] R.F. Warming, R.M. Beam and B.J. Hyett, Diagonalization and simultaneous symmetrization of the gas-dynamics matrices, Math. Comp. 29 (132) 1037-1045. · Zbl 0313.65084
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