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Affordable, entropy-consistent Euler flux functions. II: Entropy production at shocks. (English) Zbl 1280.76015
Summary: In this paper, an entropy-consistent flux is developed, continuing from the work of the previous paper [the second author, Affordable, entropy-consistent, Euler flux functions. I: Analytical results. J. Comput. Phys. (submitted)]. To achieve entropy consistency, a second and third-order differential terms are added to the entropy-conservative flux. This new flux function is tested on several one dimensional problems and compared with the original Roe flux. The new flux function exactly preserves the stationary contact discontinuity and does not capture the unphysical rarefaction shock. For steady shock problems, the new flux predicts a slightly more diffused profile whereas for unsteady cases, the captured shock is very similar to those produced by the Roe flux. The shock stability is also studied in one dimension. Unlike the original Roe flux, the new flux is completely stable which will provide as a candidate to combat multidimensional shock instability, particularly the carbuncle phenomenon.

76L05 Shock waves and blast waves in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Anderson, J.D., Modern compressible flow with historical perspective, (2004), Springer
[2] Barth, T.J., An introduction to recent developments in theory and numerics of conservation laws, Num. methods gasdynamic syst. unstructured meshes, (1999)
[3] Dumbser, M.; Moschetta, J.M.; Gressier, J., A matrix stability analysis of the carbuncle phenomenon, J. comput. phys., 197, 647, (2004) · Zbl 1079.76607
[4] Hughes, T.; Franca, L.; Mallet, M., A new finite element formulation for compressible fluid dynamics: I. symmetric forms of the compressible Euler and navier – stokes equations and the second law of thermodynamics, Comput. methods appl. mech. eng., 54, (1986) · Zbl 0572.76068
[5] F. Ismail, Toward a reliable prediction of shocks in hypersonic flow: resolving carbuncles with entropy and vorticity control, Ph.D. Thesis, The University of Michigan, 2006.
[6] K. Kitamura, P.L. Roe, F. Ismail, An evaluation of Euler fluxes for hypersonic computations, no. 2007-4465, AIAA Conference, 2007.
[7] C. Laney, Computational gasdynamics, 1998. · Zbl 0947.76001
[8] P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, in: SIAM Regional Conference Series in Applied Mathematics, 1972.
[9] Nishikawa, H.; Kitamura, K., Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers, J. comput. phys., 227, 2560-2581, (2008) · Zbl 1388.76185
[10] Osher, S., Riemann solvers, the entropy condition and difference approximations, SIAM, 21, 217-235, (1984) · Zbl 0592.65069
[11] Pandolfi, M.; D’Ambrosio, D., Numerical instabilities in upwind methods, J. comput. phys., 166, 271-301, (2001) · Zbl 0990.76051
[12] Quirk, J.J., A contribution to the great Riemann solver debate, Int. J. num. methods fluids, 18, 555-574, (1994) · Zbl 0794.76061
[13] P.L. Roe, K. Kitamura, Artificial surface tension to stabilize captured shockwaves, no. 2008-3991, AIAA Conference, 2008.
[14] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[15] Roe, P.L., Sonic flux formulae, SIAM J. sci. stat. comput., 13, 611-630, (1992) · Zbl 0747.65073
[16] P.L. Roe, Affordable, entropy-consistent, Euler flux functions I. Analytical results, J. Comput. Physics, submitted for publication.
[17] Tadmor, E.; Zhong, W., Entropy stable approximations of navier – stokes equations with no artificial numerical viscosity, J. hyperbolic DE’s, (2005)
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