Skip to main content Accessibility help
×
Home

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

  • Ee Han (a1), Jiequan Li (a2) and Huazhong Tang (a3)

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].

Copyright

Corresponding author

Corresponding author.Email:jiequan@bnu.edu.cn

References

Hide All
[1]Ben-Artzi, M., and Falcovitz, J., A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys., 55 (1984), 1–32.
[2]Ben-Artzi, M., and Falcovitz, J., Anupwind second-order scheme for compressible ductflows, SIAM J. Sci. Stat. Comput., 7 (1986), 744–768.
[3]Ben-Artzi, M., The generalized Riemann problem for reactive flows, J. Comput. Phys., 81 (1989), 70–101.
[4]Ben-Artzi, M., and Falcovitz, J., Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge University Press, 2003.
[5]Ben-Artzi, M., and Li, J., Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem, Numer. Math., 106 (2007), 369–425.
[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.
[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.
[8]Ben-Dor, G., Shock Wave Reflection Phenomena, Springer-Verlag, 1992.
[9]Brackbill, J. U., An adaptive grid with directional control, J. Comput. Phys., 108 (1993), 38–50.
[10]Brackbill, J. U., and Saltzman, J. S., Adaptive zoning for singular problems in two dimensions, J. Comput. Phys., 46 (1982), 342–368.
[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.
[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.
[13]Chang, T., and Chen, G. Q., Diffraction of planar shock along compressive corner, Acta. Math. Sci., 6 (1986), 241–257.
[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.
[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.
[16]Chen, G.-Q., and Feldman, M., Global solutions of shock reflection by large-angle wedges for potential flow, Annals. Math., to appear, 2007.
[17]Chen, S. X., Mach configuration in pseudo-stationary compressible flow, J. Am. Math. Soc., 21 (2008), 63–100.
[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.
[19]Dam, A. van, and Zegeling, P. A., Balanced monitoring of flow phenomena in moving mesh methods, Commun. Comput. Phys., 7 (2010), 138–170.
[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.
[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.
[22]Dvinsky, A. S., Adaptive grid generation from harmonic maps on Riemannian manifolds, J. Comput. Phys., 95 (1991), 450–476.
[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.
[24]Falcovitz, J., and Birman, A., A singularities tracking conservation laws scheme for compressible duct flows, J. Comput. Phys., 115 (1994), 431–439.
[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.
[26]Huang, W., Variational mesh adaptation: isotropy and equidistribution, J. Comput. Phys., 174 (2001), 903–924.
[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.
[28]Godlewski, E., and Raviart, P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996.
[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.
[30]Han, J. Q., and Tang, H. Z., An adaptive moving mesh method for multidimensional ideal magnetohydrodynamics, J. Comput. Phys., 220 (2007), 791–812.
[31]Guderly, K. G., The Theory of Transonic Flow, Pergamon Press, London, 1962.
[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.
[33]Lax, P., Computational fluid dynamics, J. Sci. Comput., 31 (2007), 185–193.
[34]Li, J., On the two-dimensional gas expansion for compressible Euler equations, SIAM J. Appl. Math., 62 (2001), 831–852.
[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.
[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.
[37]Li, J., Zhang, T., and Yang, S., The Two-dimensional Riemann Problem in Gas Dynamics, Ad-dison Wesley Longman, 1998.
[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.
[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.
[40]Neumann, J. V., Oblique reflection of shocks, Navy Department, Bureau of Ordance, Explosive Research Report, No. 12, 1943.
[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.
[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.
[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.
[44]Miller, K., and Miller, R. N., Moving finite element. I, SIAM J. Numer Anal., 18 (1981), 1019–1032.
[45]Rault, A., Chiavassa, G., and Donat, R., Shock-vortex interactions at high Mach numbers, J. Sci. Comput., 19 (2003), 347–371.
[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.
[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.
[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.
[49]Tang, H. Z., A moving mesh method for the Euler flow calculations using a directional monitor function, Commun. Comput. Phys., 1 (2006), 656–676
[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.
[51]Tang, T., Moving mesh methods for computational fluid dynamics, Contem. Math., 383 (2005), 185–218.
[52]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, 1997.
[53]Leer, B. van, Towards the ultimate conservative difference scheme V, J. Comput. Phys., 32 (1979), 101–136.
[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.
[55]Wang, D., and Wang, X., A three-dimensional adaptive method based on the iterative grid redistribution, J. Comput. Phys., 199 (2004), 423–436.
[56]Winslow, A., Numerical solution of the quasi-linear Poisson equation, J. Comput. Phys., 1 (1967), 149–172.
[57]Woodward, P., and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), 115–173.
[58]Wu, S. J., Mathematical analysis of vortex sheets, Commun. Pure. Appl. Math., 59 (2006), 1065–1206.
[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.
[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.
[61]Zheng, Y., Systems of Conservation Laws: Two-dimensional Riemann Problems, Birkhäuser, 2001.

Keywords

Related content

Powered by UNSILO

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

  • Ee Han (a1), Jiequan Li (a2) and Huazhong Tang (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.