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A Few Benchmark Test Cases for Higher-Order Euler Solvers

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  12 September 2017

Liang Pan ,
Jiequan Li  and
Kun Xu
Liang Pan*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China
Jiequan Li*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China
Kun Xu*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
*
*Corresponding author. Email addresses:[email protected] (L. Pan), [email protected] (J. Q. Li), [email protected] (K. Xu)
*Corresponding author. Email addresses:[email protected] (L. Pan), [email protected] (J. Q. Li), [email protected] (K. Xu)
*Corresponding author. Email addresses:[email protected] (L. Pan), [email protected] (J. Q. Li), [email protected] (K. Xu)
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Abstract

There have been great efforts on the development of higher-order numerical schemes for compressible Euler equations in recent decades. The traditional test cases proposed thirty years ago mostly target on the strong shock interactions, which may not be adequate enough for evaluating the performance of current higher-order schemes. In order to set up a higher standard for the development of new algorithms, in this paper we present a few benchmark cases with severe and complicated wave structures and interactions, which can be used to clearly distinguish different kinds of higher-order schemes. All tests are selected so that the numerical settings are very simple and any higher order scheme can be straightforwardly applied to these cases. The examples include highly oscillatory solutions and the large density ratio problem in one dimensional case. In two dimensions, the cases include hurricane-like solutions; interactions of planar contact discontinuities with asymptotic large Mach number (the composite of entropy wave and vortex sheets); interaction of planar rarefaction waves with transition from continuous flows to the presence of shocks; and other types of interactions of two-dimensional planar waves. To get good performance on all these cases may push algorithm developer to seek for new methodology in the design of higher-order schemes, and improve the robustness and accuracy of higher-order schemes to a new level of standard. In order to give reference solutions, the fourth-order gas-kinetic scheme (GKS) will be used to all these benchmark cases, even though the GKS solutions may not be very accurate in some cases. The main purpose of this paper is to recommend other CFD researchers to try these cases as well, and promote further development of higher-order schemes.

Keywords

Euler equationstwo-dimensional Riemann problemsfourth-order gas-kinetic schemewave interactions

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 711 - 736
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Borges, R., Carmona, M., Costa, B. and Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), pp. 3191–3211.CrossRefGoogle Scholar
[2] Acker, F., De R. Borges, R. B. and Costa, B., An improved WENO-Z scheme, J. Comput. Phys., 313 (2016), pp. 726–753.Google Scholar
[3] Ben-Artzi, M., Falcovitz, J., A second-order Godunov-type scheme for compressible uid dynamics, J. Comput. Phys., 55 (1984), pp. 1–32.CrossRefGoogle Scholar
[4] Ben-Artzi, M., Li, J. and Warnecke, G., A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys., 218 (2006), pp. 19–43.CrossRefGoogle Scholar
[5] Ben-Artzi, M. and Li, J., Hyperbolic conservation laws: Riemann invariants and the generalized Riemann problem, Numerische Mathematik, 106 (2007), pp. 369–425.Google Scholar
[6] Bhatnagar, P. L., Gross, E. P. and Krook, M., A Model for Collision Processes in Gases I: Small Amplitude Processes in Charged and Neutral One-Component Systems, Phys. Rev., 94 (1954), pp. 511–525.CrossRefGoogle Scholar
[7] Boris, J. P., A fluid transport algorithm that works, in: Computing as a Language of Physics, International Atomic Energy Commision, (1971), pp. 171–189.Google Scholar
[8] Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, 3rd ed, Cambridge University Press, (1990).Google Scholar
[9] Christlieb, A. J., Gottlieb, S., Grant, Z. and Seal, D. C., Explicit strong stability preserving multistage two-derivative time-stepping scheme, J. Sci. Comp., 68 (2016), pp. 914–942.Google Scholar
[10] Cockburn, B. and Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52 (1989), pp. 411–435.Google Scholar
[11] Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Springer, (1948).Google Scholar
[12] Cockburn, B. and Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199–224.Google Scholar
[13] Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), pp. 357–393.Google Scholar
[14] Han, E., Li, J. and Tang, H., Accuracy of the adaptive GRP scheme and the simulation of 2-D Riemann problem for compressible Euler equations, Comm. Comput. Phys., 10 (2011), pp. 577–606.Google Scholar
[15] Glimm, J., Ji, X., Li, J., Li, X., Zhang, P., Zhang, T. and Zheng, Y., Transonic shock formation in a rarefaction Riemann problem for the 2D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), pp. 720–742.CrossRefGoogle Scholar
[16] Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys., 71 (1987), pp. 231–303.CrossRefGoogle Scholar
[17] Jiang, G. S. and Shu, C. W., Efficient implementation of Weighted ENO schemes, J. Comput. Phys., 126 (1996), pp 202–228.Google Scholar
[18] Kolgan, V. P., Application of the principle of minimum values of the derivative to the construction of finite-difference schemes for calculating discontinuous solutions of gas dynamics, Scientific Notes of TsAGI, 3 (1972), pp. 68–77.Google Scholar
[19] Kreiss, H. O. and Lorenz, J., Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press, (2004).CrossRefGoogle Scholar
[20] Kurganov, A. and Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Meth. Part. Diff. Eqs., 18 (2002), pp. 584–608.Google Scholar
[21] Lax, P. D. and Liu, X. D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), pp. 319–340.Google Scholar
[22] Lax, P. and Wendroff, B., Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), pp. 217–237.Google Scholar
[23] Lee, C. B., New features of CS solitons and the formation of vortices, Phys. Lett. A, 247(6) (2008), pp. 397–402.CrossRefGoogle Scholar
[24] Lee, C. B., Possible universal transitional scenario in a flat plate boundary layer: Measurement and visualization, Phys. Rev. E, 62(3) (2000), 3659.Google Scholar
[25] Lee, C. B. and Wu, J. Z., Transition in wall-bounded flows, Appl. Mech. Rev., 61(3) (2008), 0802.CrossRefGoogle Scholar
[26] Li, J., Du, Z., A Two-Stage Fourth Order Time-Accurate Discretization for Lax-Wendroff Type Flow Solvers, I. Hyperbolic Conservation Laws, Mathematics, 2016.Google Scholar
[27] Li, J., Zhang, T. and Yang, S., The Two-Dimensional Riemann Problem in Gas Dynamics, Addison Wesley Longman, (1998).Google Scholar
[28] Li, J., Note on the compressible Euler equations with zero temperature, Appl. Math. Lett., 14 (2001), pp. 519–523.Google Scholar
[29] Li, J. and Zheng, Y., Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations, Comm. Math. Phys., 296 (2010), pp. 303–321.Google Scholar
[30] Li, Q., Xu, K. and Fu, S., A high-order gas-kinetic Navier-Stokes flow solver, J. Comput. Phys., 229 (2010), pp. 6715–6731.Google Scholar
[31] Liu, X. D., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), pp. 200–212.Google Scholar
[32] Luo, J. and Xu, K., A high-order multidimensional gas-kinetic scheme for hydrodynamic equations, Science China Technological Sciences, 56 (2013), pp. 2370–2384.Google Scholar
[33] Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Springer-Verlag, New York, 53 (1984).Google Scholar
[34] Pan, L., Xu, K., Li, Q. and Li, J., An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Navier-Stokes equations, J. Comput. Phys. 326 (2016), pp. 197–221.Google Scholar
[35] Reed, W. H. and Hill, T. R., Triangular Mesh Methods for the Neutron Transport Equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, 1973.Google Scholar
[36] W. E, , Rykov, Y. G. and Sinai, Y. G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), pp. 349–380.Google Scholar
[37] Schulz-Rinne, C. W., Collins, J. P. and Glaz, H. M., Numerical solution of the Riemann problem for twodimensional gas dynamics, SIAM J. Sci. Comput., 14 (1993), pp. 1394–1414.CrossRefGoogle Scholar
[38] Seal, D. C., Güclü, Y. and Christlieb, A. J., High-order multiderivative time integrators for hyperbolic conservation laws, J. Sci. Comp., 60 (2014), pp. 101–140.CrossRefGoogle Scholar
[39] Shi, J., Zhang, Y. T. and Shu, C. W., Resolution of high order WENO schemes for complicated flow structures, J. Comput. Phys., 186 (2003), pp. 690–696.Google Scholar
[40] Sheng, W. and Zhang, T., The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc., 654(654) (1999), pp. 77.Google Scholar
[41] Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471.CrossRefGoogle Scholar
[42] Tang, H. Z. and Liu, T. G., A note on the conservative schemes for the Euler equations, J. Comput. Phys., 218 (2006), pp. 451–459.Google Scholar
[43] Titarev, V. A. and Toro, E. F., Finite volume WENO schemes for three-dimensional conservation laws, J. Comput. Phys. 201 (2014), pp. 238–260.CrossRefGoogle Scholar
[44] Van Leer, B., Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's method, J. Comput. Phys., 32 (1979), pp. 101–136.CrossRefGoogle Scholar
[45] Woodward, P. and Colella, P., Numerical simulations of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115–173.CrossRefGoogle Scholar
[46] Xu, K., Direct modeling for computational fluid dynamics: construction and application of unfied gas kinetic schemes, World Scientific, (2015).Google Scholar
[47] Xu, K., A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), pp. 289–335.Google Scholar
[48] Zhang, T. and Zheng, Y., Exact spiral solutions of the two-dimensional Euler equations, Discrete Contin. Dynam. Systems, 3 (1997), pp. 117–133.Google Scholar
[49] Zhang, T. and Zheng, Y., Conjecture on the structure of solutions of the Riemann problem for two dimensional gas dynamics systems, SIAM J. Math. Anal., 21 (1990), pp. 593–630.Google Scholar