Abstract
Nowadays, the upwind schemes are in a rapid development to capture shock accurately. However, these upwind schemes’ properties at low speeds, such as their reconstruction scheme dependencies, grid dependencies, and Mach number dependencies, are concerned by few people. In this paper, a systematic study on their low speeds’ issues is conducted. Through a series of tests, we can find that most parameter-free upwind schemes, widely used in practice today, are not applicable to low speeds’ simulations. In contrast, SLAU and SLAU2 can give reliable results. Also, the upwind scheme’s influence on the accuracy is stronger than the reconstruction scheme’s influence at low speeds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Li X S, Gu C W. An all-speed Roe-type scheme and its asymptotic analysis of low Mach number behavior. J Comp Phys, 2008, 227: 5144–5159
Liou M S. A further development of the AUSM_Scheme towards robust and accurate solutions for all speeds. AIAA Paper, 2003
Jack R E, Liou M S. Low-diffusion flux-splitting methods for flows at all speeds. AIAA Paper, 1997
Weiss J M, Smith W A. Preconditioning applied to variable and constant density flows. AIAA J, 1995, 33: 2050–2057
Turkel E. Preconditioning technique in Computational Fluid Dynamics. Ann Rev Fluid Mech, 1999, 31: 385–416
Unrau D, Zingg D W. Viscous airfoil computations using local preconditioning. AIAA Paper, 1997
Kitamura K, Shima E, Fujimoto K, et al. Performance of low-dissipation Euler fluxes and preconditioned LU-SGS at low speeds. Comm Comput Phys, 2011, 10: 90–119
Kitamura K, Shima E. Parameter-free simple low-dissipation AUSM-family scheme for all speeds. AIAA J, 2011, 49: 1693–1709
Roe P L. Approximate Riemann solvers, parameter vectors and difference schemes. J Comp Phys, 1981, 43: 357–372
Liou M S. A sequal to AUSM: AUSM+. J Comp Phys, 1996, 129: 364–382
van Leer B. Flux vector splitting for the Euler equations. In: Eighth International Conference of Numerical Methods in Fluid Dynamics, Berlin, Germany, 1982
Kim K H, Rho O H. Methods for the accurate computations of hypersonic flows I: AUSMPW+ scheme. J Comp Phys, 2001, 174: 38–80
Kitamura K, Shima E. Towards shock-stable and accurate hypersonic heating computations: A new pressure flux for AUSM-family schemes. J Comp Phys, 2013, 245: 62–83
van Leer B. Towards the ultimate conservation difference scheme V: A second-order sequal to Godunov’s method. J Comp Phys, 1979, 32: 101–136
Kim K H, Kim C. Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows Part II: Multi-dimensional limiting process. J Comp Phys, 2005, 208: 570–615
Jiang G S, Shu C W. Efficient implementation of weighted ENO schemes. J Comp Phys, 1996, 126: 202–228
Shen Y, Zha G, Wang B. Improvement of stability and accuracy for weighted essentially non-oscillatory scheme. AIAA J, 2009, 47: 331–344
Robert H, Robert W, Pieter G. Evaluation of two high order WENO schemes. AIAA Paper, 2007
Bassi F, Rebay S. A high-order accurate finite element method for the numerical solution of the compressible Navier-Stokes equations. J Comp Phys, 2005, 208: 570–615
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Qu, F., Yan, C., Yu, J. et al. A study of parameter-free shock capturing upwind schemes on low speeds’ issues. Sci. China Technol. Sci. 57, 1183–1190 (2014). https://doi.org/10.1007/s11431-014-5547-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11431-014-5547-8