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Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

Published online by Cambridge University Press:  20 August 2015

Ee Han ,
Jiequan Li  and
Huazhong Tang
Ee Han*
Affiliation:
IAN, Department of Mathematics, Magdeburg University, D-39106, Germany
Jiequan Li*
Affiliation:
School of Mathematical Science, Beijing Normal University, Beijing 100875, China
Huazhong Tang*
Affiliation:
CAPT and LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
*
Email:eehan84@yahoo.com
Corresponding author.Email:jiequan@bnu.edu.cn
Email:hztang@math.pku.edu.cn
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Abstract

The adaptive generalized Riemann problem (GRP) scheme for 2-D compressible fluid flows has been proposed in [J. Comput. Phys., 229 (2010), 1448-1466] and it displays the capability in overcoming difficulties such as the start-up error for a single shock, and the numerical instability of the almost stationary shock. In this paper, we will provide the accuracy study and particularly show the performance in simulating 2-D complex wave configurations formulated with the 2-D Riemann problems for compressible Euler equations. For this purpose, we will first review the GRP scheme briefly when combined with the adaptive moving mesh technique and consider the accuracy of the adaptive GRP scheme via the comparison with the explicit formulae of analytic solutions of planar rarefaction waves, planar shock waves, the collapse problem of a wedge-shaped dam and the spiral formation problem. Then we simulate the full set of wave configurations in the 2-D four-wave Riemann problems for compressible Euler equations [SIAM J. Math. Anal., 21 (1990), 593-630], including the interactions of strong shocks (shock reflections), vortex-vortex and shock-vortex etc. This study combines the theoretical results with the numerical simulations, and thus demonstrates what Ami Harten observed “for computational scientists there are two kinds of truth: the truth that you prove, and the truth you see when you compute” [J. Sci. Comput., 31 (2007), 185-193].

Keywords

65M0676M1235L6065M08Adaptive GRP scheme2-D Riemann problemscollapse of a wedge-shaped damspiral formationshock reflectionsvortex-shock interaction
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 3 , September 2011 , pp. 577 - 609
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Ben-Artzi, M., and Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys., 55 (1984), 1–32.Google Scholar
[2]Ben-Artzi, M., and Falcovitz, J., Anupwind second-order scheme for compressible ductflows, SIAM J. Sci. Stat. Comput., 7 (1986), 744–768.Google Scholar
[3]Ben-Artzi, M., The generalized Riemann problem for reactive flows, J. Comput. Phys., 81 (1989), 70–101.CrossRefGoogle Scholar
[4]Ben-Artzi, M., and Falcovitz, J., Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge University Press, 2003.Google Scholar
[5]Ben-Artzi, M., and Li, J., Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem, Numer. Math., 106 (2007), 369–425.Google Scholar
[6]Ben-Artzi, M., Li, J., and Warnecke, G., A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys., 218 (2006), 19–34.Google Scholar
[7]Birman, A., Har’el, N. Y., Falcovitz, J., Ben-Artzi, M., and Feldman, U., Operator-split computation of 3-D symmetric flow, CFD J., 10 (2001), 37–43.Google Scholar
[8]Ben-Dor, G., Shock Wave Reflection Phenomena, Springer-Verlag, 1992.Google Scholar
[9]Brackbill, J. U., An adaptive grid with directional control, J. Comput. Phys., 108 (1993), 38–50.Google Scholar
[10]Brackbill, J. U., and Saltzman, J. S., Adaptive zoning for singular problems in two dimensions, J. Comput. Phys., 46 (1982), 342–368.Google Scholar
[11]Cao, W., Huang, W., and Russell, R. D., A study of monitor functions for two-dimensional adaptive mesh generation, SIAM J. Sci. Comput., 20 (1999), 1978–1999.CrossRefGoogle Scholar
[12]Cao, W., Huang, W., and Russell, R. D., An r-adaptive finite element method based upon moving mesh PDEs, J. Comput. Phys., 149 (1999), 221–244.Google Scholar
[13]Chang, T., and Chen, G. Q., Diffraction of planar shock along compressive corner, Acta. Math. Sci., 6 (1986), 241–257.Google Scholar
[14]Chang, T., and Chen, G. Q., Some fundamental concepts about system of two spatial dimensional conservation laws, Acta. Math. Sci., 6 (1986), 463–474.Google Scholar
[15]Chang, T., Chen, G. Q., and Yang, S., On the 2-D Riemann problem for the compressible Euler equations I: interaction of shocks and rarefaction waves, Dis. Cont. Dyn. Sys., 1 (1995), 555–584, II: interaction of contact discontinuities, Dis. Contin. Dyn. Syst., 6 (2000), 419–430.Google Scholar
[16]Chen, G.-Q., and Feldman, M., Global solutions of shock reflection by large-angle wedges for potential flow, Annals. Math., to appear, 2007.Google Scholar
[17]Chen, S. X., Mach configuration in pseudo-stationary compressible flow, J. Am. Math. Soc., 21 (2008), 63–100.Google Scholar
[18]Dam, A. van, and Zegeling, P. A., A robust moving mesh finite volume method applied to 1-D hyperbolic conservation laws from magnetohydrodynamics, J. Comput. Phys., 216 (2006), 526–546.Google Scholar
[19]Dam, A. van, and Zegeling, P. A., Balanced monitoring of flow phenomena in moving mesh methods, Commun. Comput. Phys., 7 (2010), 138–170.Google Scholar
[20]Davis, S. F., and Flaherty, J. E., An adaptive finite element method for initial-boundary value problems for partial differential equations, SIAM J. Sci. Stat. Comput., 3 (1982), 6–27.Google Scholar
[21]Di, Y., Li, R., Tang, T., and Zhang, P. W., Moving mesh finite element methods for the incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 26 (2005), 1036–1056.Google Scholar
[22]Dvinsky, A. S., Adaptive grid generation from harmonic maps on Riemannian manifolds, J. Comput. Phys., 95 (1991), 450–476.Google Scholar
[23]Falcovitz, J., Alfandary, G., and Hanoch, G., A 2-D conservation laws scheme for compressible flows with moving boundaries, J. Comput. Phys., 138 (1997), 83–102.Google Scholar
[24]Falcovitz, J., and Birman, A., A singularities tracking conservation laws scheme for compressible duct flows, J. Comput. Phys., 115 (1994), 431–439.Google Scholar
[25]Han, E., Li, J. Q., and Tang, H. Z., An adaptive GRP scheme for compressible fluid flows, J. Comput. Phys., 229 (2010), 1448–1466.Google Scholar
[26]Huang, W., Variational mesh adaptation: isotropy and equidistribution, J. Comput. Phys., 174 (2001), 903–924.Google Scholar
[27]Glimm, J., Ji, X., Li, J., Li, X., Zhang, T., Zhang, P., and Zheng, Y., Transonic shock formation in a rarefaction Riemann problem for the 2-D compressible Euler equations, SIAM J. Appl. Math., 69 (2008), 720–742.CrossRefGoogle Scholar
[28]Godlewski, E., and Raviart, P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996.Google Scholar
[29]Godunov, S. K., A finite difference method for the numerical computation and disontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47 (1959), 271–295.Google Scholar
[30]Han, J. Q., and Tang, H. Z., An adaptive moving mesh method for multidimensional ideal magnetohydrodynamics, J. Comput. Phys., 220 (2007), 791–812.Google Scholar
[31]Guderly, K. G., The Theory of Transonic Flow, Pergamon Press, London, 1962.Google Scholar
[32]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), 584–608.Google Scholar
[33]Lax, P., Computational fluid dynamics, J. Sci. Comput., 31 (2007), 185–193.Google Scholar
[34]Li, J., On the two-dimensional gas expansion for compressible Euler equations, SIAM J. Appl. Math., 62 (2001), 831–852.Google Scholar
[35]Li, J., and Chen, G., The generalized Riemann problem method for the shallow water equations with bottom topography, Int. J. Numer. Methods. Eng., 65 (2006), 834–862.Google Scholar
[36]Li, J., Liu, T., and Sun, Z., Implementation of the GRP scheme for computing radially symmetric compressible fluid flows, J. Comput. Phys., 228 (2009), 5867–5887.Google Scholar
[37]Li, J., Zhang, T., and Yang, S., The Two-dimensional Riemann Problem in Gas Dynamics, Ad-dison Wesley Longman, 1998.Google Scholar
[38]Li, J., and Zheng, Y., Interaction of rarefaction waves of the two-dimensional self-similar Euler equations, Arch. Rat. Mech. Anal., 193 (2009), 623–657.Google Scholar
[39]Li, J., and Zheng, Y., Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations, Commun. Math. Phys., 296 (2010), 303–326.Google Scholar
[40]Neumann, J. V., Oblique reflection of shocks, Navy Department, Bureau of Ordance, Explosive Research Report, No. 12, 1943.Google Scholar
[41]Li, R., Tang, T., and Zhang, P. W., Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys., 170 (2001), 562–588.Google Scholar
[42]Li, R., Tang, T., and Zhang, P. W., A moving mesh finite element algorithm for singular problems in two and three space dimensions, J. Comput. Phys., 177 (2002), 365–393.CrossRefGoogle Scholar
[43]Liu, X. D., and Lax, P. D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319–340.Google Scholar
[44]Miller, K., and Miller, R. N., Moving finite element. I, SIAM J. Numer Anal., 18 (1981), 1019–1032.Google Scholar
[45]Rault, A., Chiavassa, G., and Donat, R., Shock-vortex interactions at high Mach numbers, J. Sci. Comput., 19 (2003), 347–371.Google Scholar
[46]Schulz-Rinne, C. W., Collins, J. P., and Glaz, H. M., Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. Sci. Comput., 14 (1993), 1394–1414.Google Scholar
[47]Sheng, W. C., and Yin, G., Transonic shock and supersonic shock in the regular reflection of a planar shock, Z. Angew. Math. Phys., 60 (2009), 438–449.Google Scholar
[48]Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27 (1978), 1–31.Google Scholar
[49]Tang, H. Z., A moving mesh method for the Euler flow calculations using a directional monitor function, Commun. Comput. Phys., 1 (2006), 656–676Google Scholar
[50]Tang, H. Z., and Tang, T., Adaptive mesh methods for one- and two- dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41 (2003), 487–515.Google Scholar
[51]Tang, T., Moving mesh methods for computational fluid dynamics, Contem. Math., 383 (2005), 185–218.Google Scholar
[52]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, 1997.Google Scholar
[53]Leer, B. van, Towards the ultimate conservative difference scheme V, J. Comput. Phys., 32 (1979), 101–136.Google Scholar
[54]Wang, C. W., Tang, H. Z., and Liu, T. G., An adaptive ghost fluid finite volume method for compressible gas-water simulations, J. Comput. Phys., 227 (2008), 6385–6409.Google Scholar
[55]Wang, D., and Wang, X., A three-dimensional adaptive method based on the iterative grid redistribution, J. Comput. Phys., 199 (2004), 423–436.Google Scholar
[56]Winslow, A., Numerical solution of the quasi-linear Poisson equation, J. Comput. Phys., 1 (1967), 149–172.Google Scholar
[57]Woodward, P., and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), 115–173.Google Scholar
[58]Wu, S. J., Mathematical analysis of vortex sheets, Commun. Pure. Appl. Math., 59 (2006), 1065–1206.Google Scholar
[59]Zegeling, P. A., Boer, W. D. de, and Tang, H. Z., Robust and efficient adaptive moving mesh solution of 2-D Euler equation, Contem. Math., 383 (2005), 375–386.Google Scholar
[60]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), 593–630.Google Scholar
[61]Zheng, Y., Systems of Conservation Laws: Two-dimensional Riemann Problems, Birkhäuser, 2001.Google Scholar