Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T07:59:57.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Part of: Shock waves and blast waves Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 February 2016

Jianming Liu ,
Jianxian Qiu ,
Mikhail Goman ,
Xinkai Li  and
Meilin Liu
Jianming Liu
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China Faculty of Technology, De Montfort University, Leicester LE1 9BH, England
Jianxian Qiu*
Affiliation:
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen 361005, China
Mikhail Goman
Affiliation:
Faculty of Technology, De Montfort University, Leicester LE1 9BH, England
Xinkai Li
Affiliation:
Faculty of Technology, De Montfort University, Leicester LE1 9BH, England
Meilin Liu
Affiliation:
Shanghai Institute of Satellite Engineering, Shanghai 200240, China
*
*Corresponding author. Email address: jxqiu@xmu.edu.cn (J.-X. Qiu)
Get access

Abstract

In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

Keywords

Discontinuous Galerkin methodadaptive Cartesian gridpositivity-preservingimmersed boundary methodcomplex geometry

MSC classification

Secondary: 65M50: Mesh generation and refinement 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76L05: Shock waves and blast waves
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 1 , February 2016 , pp. 87 - 110
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Yang, G., Causon, D. M., Ingram, D. M., Saunders, R., and Batten, P., A Cartesian cut cell method for compressible flows Part A: static body problems, Aeronaut. J., vol. 101 (1997), pp. 47–56.CrossRefGoogle Scholar
[2]Sjögreen, B. and Petersson, N, A Cartesian embedded boundary method for hyperbolic conservation laws, Commun. Comput. Phys., vol. 2 (2007), pp. 1199–1219.Google Scholar
[3]Forrer, H. and Jeltsch, R., A higher-order boundary treatment for Cartesian-grid methods, J. Comput. Phys., vol. 140 (1998), pp. 259–277.CrossRefGoogle Scholar
[4]Dadone, A. A and Grossman, B., Ghost-cell method for inviscid two-dimensional flows on Cartesian grids, AIAA J., vol. 42 (2004), pp. 2499–2507.CrossRefGoogle Scholar
[5]Liu, J. M., Zhao, N., and Hu, O., The ghost cell method and its applications for inviscid compressible flow on adaptive tree Cartesian grids, Adv. Appl. Math. Mech., vol. 1 (2009), pp. 664–82.CrossRefGoogle Scholar
[6]Ji, H., Lien, F. S., and Yee, E., A robust and efficient hybrid cut-cell/ghost-cell method with adaptive mesh refinement for moving boundaries on irregular domains, Comput. Methods Appl. Mech. Engrg, 198 (2008), pp. 432–48.CrossRefGoogle Scholar
[7]Berger, M. J., Helzel, C., and Leveque, R. J., H-box methods for the approximation of hyperbolic conservation laws on irregular grids, SIAM J. Numer. Anal., vol. 41 (2003), pp.893–918.CrossRefGoogle Scholar
[8]Helzel, C., Berger, M. J., and Leveque, R. J., A high-resolution rotated grid method for conservation laws with embedded geometries, SIAM J. Sci. Comput., vol. 26 (2005), pp.785–809.CrossRefGoogle Scholar
[9]Colella, P., Graves, D. T., Keen, B. J., and Modian, D., A Cartesian grid embedded boundary method for hyperbolic conservation laws, J. Comput. Phys., vol. 211 (2006), pp.347–366.CrossRefGoogle Scholar
[10]Sambasivan, S. K and Udaykumar, H. S., Ghost fluid method for strong shockinter actions Part 2: Immersed solid boundaries, AIAA J. vol. 47 (2009), pp. 2923–2937.CrossRefGoogle Scholar
[11]Chaudhuri, A., Hadjadj, A., and Chinnayya, A., On the use of immersed boundary methods for shock/obstacle interactions, J. Comput. Phys., vol. 230 (2011), pp. 1731–1748.CrossRefGoogle Scholar
[12]Liu, J. M., Qiu, J. X., Zhao, N., Hu, O., Goman, M., and Li, X. K., Adaptive Runge-Kutta discontinuous Galerkin method for complex geometry problems on Cartesian grid, Int. J. Numer. Meth. Fluids, vol. 73 (2013), pp. 847–868.CrossRefGoogle Scholar
[13]Zhang, X. and Shu, C.-W, On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., vol.229 (2010), pp. 8918–8934.CrossRefGoogle Scholar
[14]Zhang, X. and Shu, C.-W, Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms, J. Comput. Phys., vol. 230 (2011), pp.1238–1248.CrossRefGoogle Scholar
[15]Zhang, X., Xia, Y., and Shu, C.-W., Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. Comput., vol. 50 (2012), pp. 29–62.CrossRefGoogle Scholar
[16]Wang, C., Zhang, X., Shu, C.-W., and Ning, J. G., Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations, J. Comput. Phys., vol. 231 (2012), pp. 653–665.CrossRefGoogle Scholar
[17]Kotov, D. V., Yee, H. C., and Sjögreen, B., Comparative study on high-order positivity-preserving WENO schemes, ARC-E-DAA-TN8768, SIAM Conference on Numerical Combustion, San Antonio, TX, USA, April 8th-10th, 2013.Google Scholar
[18]Kontzialis, K. and Ekaterinaris, J. A., High order discontinuous Galerkin discretizations with a new limiting approach and positivity preservation for strong moving shocks, Comput Fluids, vol. 71 (2013), pp. 98–112.CrossRefGoogle Scholar
[19]Kim, S. D., Lee, B. J., Lee, H. J., and Jeung, I. S., Robust HLLC Riemann solver with weighted average flux scheme for strong shock, J. Comput. Phys., vol. 228 (2009), pp.7634–7642.CrossRefGoogle Scholar
[20]Cockburn, B. and Shu, C.-W, The Runge-Kutta local projection P1-discontinuous finite element method for scalar conservation laws, Math. Model Numer. Anal., vol. 25 (1991), pp. 337–361.CrossRefGoogle Scholar
[21]Cockburn, B. and Shu, C.-W, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., vol. 52 (1989), pp. 411–435.Google Scholar
[22]Cockburn, B. and Shu, C.-W, The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems, J. Comput. Phys., vol. 141 (1998), pp. 199–224.CrossRefGoogle Scholar
[23]Cockburn, B., Lin, S. Y., and Shu, C.-W, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems, J. Comput. Phys., vol. 84 (1989), pp. 90–113.CrossRefGoogle Scholar
[24]Cockburn, B., Hou, S., and Shu, C.-W, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comp., vol. 54 (1990), pp. 545–581.Google Scholar
[25]Shu, C.-W and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., vol. 77 (1988), pp. 439–471.CrossRefGoogle Scholar
[26]Berg, M., Cheong, O., Kreveld, M., and Overmars, M., Computational Geometry: Algorithms and Applications, Springer-Verlag, Berlin Heidelberg, 2008.CrossRefGoogle Scholar
[27]Cecil, T., Osher, S., and Qian, J., Essentially non-oscillatory adaptive tree methods, J. Sci. Comput., vol. 35 (2008), pp. 25–41.CrossRefGoogle Scholar
[28]Zhu, H. Q. and Qiu, J. X., An h-adaptive RKDG method with troubled-cell indicator for two-dimensional hyperbolic conservation laws, Adv. Comput. Math., vol. 39 (2013), pp. 445–463.CrossRefGoogle Scholar
[29]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Berlin Heidelberg, Springer, 1999.CrossRefGoogle Scholar
[30]Toro, E. F., Spruce, M., and Speares, W., Restoration of the contact surface in the HLL Riemann solver, Shock Waves, vol. 4 (1994), pp. 25–34.CrossRefGoogle Scholar
[31]Harten, A., Lax, P. D., and Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., vol. 25 (1983), pp. 35–61.CrossRefGoogle Scholar
[32]Batten, P., Clarke, N., Lambert, C., and Causon, D. M., On the choice ofwavespeedsfor the HLLC Riemann solver, SIAM J. Sci. Comput., vol. 18 (1997), pp. 1553–1570.CrossRefGoogle Scholar
[33]Wang, D. H., Zhao, N., Hu, O., and Liu, J. M., A ghost fluid based front tracking method for multimedium compressible flows, Acta. Math. Sci., vol. 29 (2009), pp. 1629–1646.Google Scholar
[34]Krivodonova, L., Xin, J., Remacle, J. F., Chevaugeon, N., and Flaherty, J. E., Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math., vol. 48 (2004), pp. 323–338.CrossRefGoogle Scholar
[35]De Zeeuw, D., A Quadtree-Based Adaptively-Refined Cartesian-Grid Algorithm for Solution of the Euler Equations, PhD Thesis, University of Michigan, 1993.Google Scholar
[36]Hu, X. Y., Adams, N. A., and Shu, C.-W, Positivity-preserving method for high-order conservative schemes solving compressible Euler equations, J. Comput. Phys., vol. 242 (2013), pp. 169–180.CrossRefGoogle Scholar
[37]Lee, J. and Ruffin, S. M., Development of a Turbulent Wall-Function Based Viscous Cartesian-Grid Methodology, AIAA Paper 07-1326. 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2007.CrossRefGoogle Scholar