Skip to main content

Runge-Kutta Discontinuous Galerkin Method with Front Tracking Method for Solving the Compressible Two-Medium Flow on Unstructured Meshes

  • Haitian Lu (a1), Jun Zhu (a2), Chunwu Wang (a2) and Ning Zhao (a1)

In this paper, we extend using the Runge-Kutta discontinuous Galerkin method together with the front tracking method to simulate the compressible two-medium flow on unstructured meshes. A Riemann problem is constructed in the normal direction in the material interfacial region, with the goal of obtaining a compact, robust and efficient procedure to track the explicit sharp interface precisely. Extensive numerical tests including the gas-gas and gas-liquid flows are provided to show the proposed methodologies possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow and the interfacial vicinities of the two-medium flow in many occasions.

Corresponding author
*Corresponding author. Email: (H. T. Lu), (J. Zhu), (C. W. Wang), (N. Zhao)
Hide All
[1] Abgrall, R., How to prevent pressure oscillations in multicomponent flow calculations: a quasi-conservative approach, J. Comput. Phys., 125 (1996), pp. 150160.
[2] Caiden, R., Fedkiw, R. P. and Anderson, C., A numerical method for two-phase flow consisting of separate compressible and incompressible regions, J. Comput. Phys., 166 (2001), pp. 127.
[3] Chern, I.-L., Glimm, J., McBryan, O., Plohr, B. and Yaniv, S., Front tracking for gas dynamics, J. Comput. Phys., 62 (1986), pp. 83110.
[4] Cocchi, J.-P. and Saurel, R., A Riemann problem based method for the resolution of compressible multimaterial flows, J. Comput. Phys., 137 (1997), pp. 265298.
[5] 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. Comput., 54 (1990), pp. 545581.
[6] 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., 84 (1989), pp. 90113.
[7] 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. 411435.
[8] Cockburn, B. and Shu, C.-W., The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Math. Model. Numer. Anal., 25 (1991), pp. 337361.
[9] Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199224.
[10] Fedkiw, R. P., Coupling an Eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method, J. Comput. Phys., 175 (2002), pp. 200224.
[11] Fedkiw, R. P., Aslam, T., Merriman, B. and Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152 (1999), pp. 457492.
[12] Glimm, J., Grove, J.W., Li, X. L., Shyue, K.-M., Zeng, Y. and Zhang, Q., Three-dimensional front tracking, SIAM J. Sci. Comput., 19 (1998), pp. 703727.
[13] Glimm, J., Grove, J. W., Li, X. L. and Zhao, N., Simple front tracking, Contemp. Math., 238 (1999), pp. 133149.
[14] Glimm, J., Grove, J. W., Li, X. L., Oh, W. and Sharp, D. H., A critical analysis of Rayleigh-Taylor growth rates, J. Comput. Phys., 169 (2001), pp. 652677.
[15] Hao, Y. and Prosperetti, A., A numerical method for three-dimensional gas-liquid flow computations, J. Comput. Phys., 196 (2004), pp. 126144.
[16] Hass, J.-F. and Sturtevant, B., Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities, J. Fluid Mech., 181 (1987), pp. 4176.
[17] Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. Comput. phys., 112 (1994), pp. 3143.
[18] Larrouturou, B., How to preserve the mass fractions positivity when computing compressible multi-component flows, J. Comput. Phys., 95 (1991), pp. 5984.
[19] Liu, T. G., Khoo, B. C. and Yeo, K. S., The simulation of compressible multi-medium flow I: a new methodology with test applications to 1D gas-gas and gas-water cases, Comput. Fluids, 30 (2001), pp. 291314.
[20] Lu, H., Zhao, N. and Wang, D., A front tracking method for the simulation of compressible multimedium flows, Commun. Comput. Phys., 19 (2016), pp. 124142.
[21] Mulder, W., Osher, S. and Sethian, J. A., Computing interface motion in compressible gas dynamics, J. Comput. Phys., 100 (1992), pp. 209228.
[22] Nourgaliev, R. R., Dinh, T. N. and Theofanous, T. G., Adaptive characteristics-based matching for compressible multifluid dynamics, J. Comput. Phys., 213 (2006), pp. 500529.
[23] Osher, S. and Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169 (2001), pp. 463502.
[24] Picone, J. M. and Boris, J. P., Vorticity generation by shock propagation through bubbles in a gas, J. Fluid Mech., 189 (1988), pp. 2351.
[25] Qiu, J., Liu, T. G. and Khoo, B. C., Simulations of compressible two-medium flow by Runge-Kutta discontinuous Galerkin methods with the ghost fluid method, Commun. Comput. Phys., 3 (2008), pp. 479504.
[26] Quirk, J. J. and Karni, S., On the dynamics of a Shock-bubble interaction, J. Fluid Mech., 318 (1996), pp. 129163.
[27] Reed, W. H. and Hill, T. R., Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973.
[28] Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. Comput., 49 (1987), pp. 105121.
[29] Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77 (1988), pp. 439471.
[30] Shyue, K.-M., An efficient shock-capturing algorithm for compressible multicomponent problems, J. Comput. Phys., 142 (1998), pp. 208242.
[31] Terashima, H. and Tryggvason, G., A front tracking/ghost fluid method for fluid interfaces in compressible flows, J. Comput. Phys., 228 (2009), pp. 40124037.
[32] Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. and Jan, Y.-J., A front tracking method for the computations of multiphase flow, J. Comput. Phys., 169 (2001), pp. 708759.
[33] Wang, C. W., Liu, T. G. and Khoo, B. C., A real ghost fluid method for the simulation of multimedium compressible flow, SIAM J. Sci. Comput., 28 (2006), pp. 278302.
[34] Zhu, J. and Qiu, J., Adaptive Runge-Kutta discontinuous Galerkin methods with modified ghost fluid method for simulating the compressible two-medium flow, J. Math. Study, 47 (2014), pp. 250273.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 102 *
Loading metrics...

Abstract views

Total abstract views: 436 *
Loading metrics...

* Views captured on Cambridge Core between 11th October 2016 - 21st August 2018. This data will be updated every 24 hours.