×

On the sonic point glitch. (English) Zbl 1061.76048

Summary: This paper presents theoretical and numerical analyses of the sonic point glitch based on some numerical schemes for Burgers’ equation and Euler equations in fluid mechanics. The sonic glitch is formed in the sonic rarefaction fan. It has no any direct connection with the violation of the entropy condition or the size of numerical viscosity of a finite difference scheme. Our results show that it is mainly coming from a disparity in wave speeds across the sonic point. If numerical viscosity depends on the characteristic direction, then the disparity may be formed between the numerical and physical wave speeds around the sonic point, and triggers the sonic wiggle in numerical solution. We also find that the initial data reconstruction technique of van Leer can effectively eliminate the flaw around the sonic point for the Burgers’ equation. Some other possible cures are also suggested.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
76J20 Supersonic flows

Software:

WIGGLE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, J. D., Modern Compressible Flow with Historical Perspective (1990), McGraw-Hill: McGraw-Hill New York
[2] Davis, S. F., A simplified TVD finite difference scheme via artificial viscosity, SIAM J. Sci. Stat. Comput., 8, 1-18 (1987) · Zbl 0689.65058
[3] Engquist, B.; Osher, S., One sided difference approximations for nonlinear conservation laws, Math. Comp., 36, 321-351 (1981) · Zbl 0469.65067
[4] Godunov, S. K., A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47, 271-306 (1959) · Zbl 0171.46204
[5] Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49, 357-393 (1983) · Zbl 0565.65050
[6] Hirsch, C., Numerical Computation of Internal and External Flows, vols. 1 and 2 (1990), Wiley: Wiley New York
[7] Hui, W. H.; Kudriakov, S., On wall overheating and other computational difficulties of shock-capturing methods, CFD J., 10, 192-209 (2001)
[8] Liou, M. S.; Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107, 23-39 (1993) · Zbl 0779.76056
[9] F.J. Liu, W.W. Liou, A new approach for eliminating numerical oscillations of Roe family of schemes at sonic point, AIAA 99-0301, in: 37th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1999; F.J. Liu, W.W. Liou, A new approach for eliminating numerical oscillations of Roe family of schemes at sonic point, AIAA 99-0301, in: 37th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1999
[10] J.M. Moschetta, J. Gressier, The sonic point glitch problems: a numerical solution, in: Charles-Henri Bruneau (Ed.), 16th International Conference on Numerical Methods in Fluid Dynamics, 1998, pp. 403-408; J.M. Moschetta, J. Gressier, The sonic point glitch problems: a numerical solution, in: Charles-Henri Bruneau (Ed.), 16th International Conference on Numerical Methods in Fluid Dynamics, 1998, pp. 403-408
[11] Moschetta, J. M.; Gressier, J., A cure for the sonic point glitch, Int. J. Comput. Fluid Dynamics, 13, 143-159 (2000) · Zbl 0983.76060
[12] Moschetta, J.-M.; Pullin, D. I., A robust diffusive kinetic scheme for the Navier-Stokes/Euler equations, J. Comput. Phys., 133, 193-204 (1997) · Zbl 0882.76066
[13] Quirk, J. J., A contribution to the great Riemann solver debate, Int. J. Numer. Methods Fluids, 18, 555-574 (1994) · Zbl 0794.76061
[14] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Math. Comp., 38, 339-374 (1982) · Zbl 0483.65055
[15] Osher, S.; Chakravarthy, S., Upwind schemes and boundary conditions with applications to Euler equations in general geometries, J. Comput. Phys., 50, 447-481 (1983) · Zbl 0518.76060
[16] Pullin, D. I., Direct simulation methods for compressible inviscid ideal gas flow, J. Comput. Phys., 34, 231-244 (1980) · Zbl 0419.76049
[17] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[18] Roe, P. L., Sonic flux formulae, SIAM J. Sci. Stat. Comput., 13, 611-630 (1992) · Zbl 0747.65073
[19] Sanders, R.; Prendergast, K., The possible relation of the three-kiloparsec arm to explosions in the galactic nucleus, Astrophys. J., 188, 489-500 (1974)
[20] Steger, J. L.; Warming, R. F., Flux vector-splitting of the inviscid gas dynamic equations with applications to finite difference methods, J. Comput. Phys., 40, 263-293 (1981) · Zbl 0468.76066
[21] Tang, H. Z.; Tang, T., Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41, 487-515 (2003) · Zbl 1052.65079
[22] Tang, H. Z.; Xu, K., Pseudoparticle representation and positivity analysis of explicit and implicit Steger-Warming FVS schemes, Z. Angew. Math. Phys., 52, 847-858 (2001) · Zbl 1004.76072
[23] H.Z. Tang, K. Xu, An explanation for the sonic point glitch, preprint, 2000. Available from: <http://www.math.ntnu.no/conservation/; H.Z. Tang, K. Xu, An explanation for the sonic point glitch, preprint, 2000. Available from: <http://www.math.ntnu.no/conservation/
[24] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1999), Springer: Springer Berlin · Zbl 0923.76004
[25] van Leer, B., Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136 (1979) · Zbl 1364.65223
[26] B. van Leer, Flux-vector splitting for the Euler equations, Technical Report ICASE 82-30, NASA Langley Research Center, USA, 1982. Also in Proceedings of the 8th International Conference on Numerical Methods in Fluid Dynamics, Springer, Berlin, 1982, pp. 507-512; B. van Leer, Flux-vector splitting for the Euler equations, Technical Report ICASE 82-30, NASA Langley Research Center, USA, 1982. Also in Proceedings of the 8th International Conference on Numerical Methods in Fluid Dynamics, Springer, Berlin, 1982, pp. 507-512
[27] van Leer, B., On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Stat. Comput., 5, 1-20 (1984) · Zbl 0547.65065
[28] B. van Leer, W.T. Lee, K.G. Powell, Sonic-point capturing, AIAA-89-1945-CP, in: AIAA 9th Computational Fluid Dynamics Conference, Buffalo, NY, 1989; B. van Leer, W.T. Lee, K.G. Powell, Sonic-point capturing, AIAA-89-1945-CP, in: AIAA 9th Computational Fluid Dynamics Conference, Buffalo, NY, 1989
[29] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173 (1984) · Zbl 0573.76057
[30] K. Xu, Gas-Kinetic Schemes for Unsteady Compressible Flow Simulations, VKI Fluid Dynamics Lecture Series, 1998-03, 1998. Available from: <http://www.math.ust.hk/ makxu; K. Xu, Gas-Kinetic Schemes for Unsteady Compressible Flow Simulations, VKI Fluid Dynamics Lecture Series, 1998-03, 1998. Available from: <http://www.math.ust.hk/ makxu
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.