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An all-speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour. (English) Zbl 1388.76207
Summary: A new scheme, All-Speed-Roe scheme, was proposed for all speed flows. Compared with traditional preconditioned Roe scheme, All-Speed-Roe scheme changes non-linear eigenvalues in the numerical dissipation terms of Roe-type schemes. With an asymptotic analysis, the low Mach number behaviour of the scheme is studied theoretically in two ways. In one way, All-Speed-Roe scheme is regarded as finite magnification of Low-Speed-Roe scheme in the low Mach number limit. In the other way, a general form of All-Speed-Roe scheme is analyzed. Both ways demonstrate that All-Speed-Roe scheme has the same low Mach number behaviour as the original governing equation in the continuous case, which includes three important features: pressure variation scales with the square of the Mach number, the zero order velocity field is subject to a divergence constraint, and the second order pressure satisfies a Poisson-type equation in the case of constant-entropy. The analysis also leads to an unexpected conclusion that the velocity filed computed by traditional preconditioned Roe scheme does not satisfy the divergence constraint as the Mach number vanishes. Moreover, the analysis explains the reason of checkerboard decoupling and shows that momentum interpolation method provides a similar mechanism as traditional preconditioned Roe scheme inherently possesses to suppress checkerboard decoupling. In the end, general rulers for modifying non-linear eigenvalues are obtained. Finally, several numerical experiments are provided to support the theoretical analysis. All-Speed-Roe scheme has a sound foundation and is expected to be widely studied and applied to all speed flow calculations.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics
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##### References:
 [1] Weiss, J.M.; Smith, W.A., Preconditioning applied to variable and constant density flows, AIAA journal, 33, 2050-2057, (1995) · Zbl 0849.76072 [2] Edwards, J.R.; Liou, M.S., Low-diffusion flux-splitting methods for flows at all speeds, AIAA journal, 36, 1610-1617, (1998) [3] Turkel, E., Preconditioning techniques in computational fluid dynamics, Annual reviews of fluid mechanics, 31, 385-416, (1999) [4] Mary, I.; Sagaut, P.; Deville, M., An algorithm for unsteady viscous flows at all speeds, International journal for numerical methods in fluids, 34, 371-401, (2000) · Zbl 1003.76057 [5] Guillard, H.; Viozat, C., On the behaviour of upwind schemes in the low Mach number limit, Computers and fluids, 28, 63-86, (1999) · Zbl 0963.76062 [6] Guillard, H.; Murrone, A., On the behaviour of upwind schemes in the low Mach number limit: II. Godunov type schemes, Computers and fluids, 33, 655-675, (2004) · Zbl 1049.76040 [7] Birken, P.; Meister, A., Stability of preconditioned finite volume schemes at low Mach numbers, Numerical mathematics, 45, 463-480, (2005) · Zbl 1124.76038 [8] X.S. Li, C.W. Gu, J.Z. Xu, Research of preconditioned roe scheme for all speed flows in turbomachinery, in: ISABE Conference of AIAA, 2007-1113. [9] Klainerman, S.; Majda, A., Compressible and incompressible fluids, Communications on pure applied mathematics, 35, 629-651, (1982) · Zbl 0478.76091 [10] Klein, R., Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: one-dimensional flow, Journal of computational physics, 121, 213-237, (1995) · Zbl 0842.76053 [11] Meister, A., Asymptotic based preconditioning technique for low Mach number flows, Zeitschrift fur angewandte Mathematik und mechanik, 83, 1, 3-25, (2003) · Zbl 1010.76062 [12] Rhie, C.M.; Chow, W.L., Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA journal, 21, 1525-1532, (1983) · Zbl 0528.76044 [13] Mary, I.; Sagaut, P., Large eddy simulation of flow around an airfoil near stall, AIAA journal, 40, 1139-1145, (2002) [14] Darmofal, D.L.; Schmid, P.J., The importance of eigenvectors for local preconditioners of the Euler equations, Journal of computational physics, 127, 346-362, (1996) · Zbl 0860.76054 [15] X.S. Li, Large Eddy Simulation Based on the Computational Method of Compressible Flows and its Application to Engineering, Ph.D. Dissertation, Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, March 2006. [16] X.S. Li, J.Z. Xu, C.W. Gu, Preconditioning method and engineering application of large eddy simulation. Science in China Series G: Physics, Mechanics & Astronomy (2008), doi:10.1007/s11433-008-0054-1. [17] Lee, S.H., Cancellation problem of preconditioning method at low Mach numbers, Journal of computational physics, 225, 1199-1210, (2007) · Zbl 1343.76047
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