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An Arbitrary-Lagrangian-Eulerian High-Order Gas-Kinetic Scheme for Three-Dimensional Computations

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In the previous study [J. Comput. Phys. 417 (2020) 109558], under arbitrary-Lagrangian-Eulerian (ALE) formulation a high-order gas-kinetic scheme has been developed for the computation of two-dimensional flows. For the three-dimensional flows, due to the distorted mesh, it becomes more difficult to develop robust high-order ALE methods with the precise preservation of geometric conservation law. In this paper, the high-order gas-kinetic ALE scheme will be constructed for three-dimensional flows. The key ingredients of the scheme are the use of weighted essentially non-oscillatory (WENO) scheme for spatial reconstruction and the two-stage fourth-order discretization for temporal evolution. In the ALE formulation, in order to release the problems associated with mesh distortion and non-coplanar vertexes of a control volume, in the spatial reconstruction the selection of candidate stencils and the topologically independent linear weights have to be carefully designed. In the surface integrals for the flux transport, a bilinear interpolation is used to parameterize both grid coordinates and grid moving velocity with the preservation of the geometric conservation law. In the computation, the grid velocity is determined by the variational formulation based Lagrangian nodal solver. Numerical examples are presented to evaluate the accuracy, robustness, and the preservation of geometric conservation law of the current scheme.

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References

  1. Anderson, R.W., Dobrev, V.A., Kolev, T.V., Rieben, R.N., Tomov, V.Z.: High-order multi-material ale hydrodynamics. SIAM J. Sci. Comput. 40, B32–B58 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Barlow, A.J., Maire, P.H., Rider, W.J., Rieben, R.N., Shashkov, M.J.: Arbitrary Lagrangian Eulerian methods for modeling high-speed compressible multimaterial flows. J. Comput. Phys. 322, 603–665 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Benson, D.J.: Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Methods Appl. Mech. Eng. 99, 235–394 (1992)

    MathSciNet  MATH  Google Scholar 

  4. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)

    MATH  Google Scholar 

  5. Boscheri, W., Dumbser, M.: Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Commun. Comput. Phys. 14, 1174–1206 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Boscheri, W., Dumbser, M.: A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D. J. Comput. Phys. 275, 484–523 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Boscheri, W., Balsara, D.S.: High order direct Arbitrary-Lagrangian-Eulerian (ALE) \(P_NP_M\) schemes with WENO Adaptive-Order reconstruction on unstructured meshes. J. Comput. Phys. 398, 108899 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Boscheri, W.: Dimarco, high order central WENO-Implicit-Explicit Runge Kutta schemes for the BGK model on general polygonal meshes. J. Comput. Phys. 422, 109766 (2020)

    MathSciNet  Google Scholar 

  9. Brackbill, J.U., Saltzman, J.S.: Adaptive zoning for singular problems in two dimensions. J. Comput. Phys. 46, 342–368 (1982)

    MathSciNet  MATH  Google Scholar 

  10. Burton, D.E.: Exact Conservation of Energy and Momentum in Staggered-grid Hydrodynamics with Arbitrary Connectivity. Advances in the Free Lagrange Method. Springer, New York (1990)

    Google Scholar 

  11. Burton, D.E.: Multidimensional discretization of conservation laws for unstructured polyhedral grids. Technical Report UCRL-JC-118306, Lawrence Livermore National Laboratory (1990)

  12. Caramana, E.J., Rousculp, C.L., Burton, D.E.: A compatible, energy and symmetry preserving Lagrangian hydrodynamics algorithm in three-dimensional Cartesian geometry. J. Comput. Phys. 157, 89–119 (2000)

    MATH  Google Scholar 

  13. Caramana, E.J., Shashkov, M.J., Whalen, P.P.: Formulations of artificial viscosity for multi-dimensional shock wave computations. J. Comput. Phys. 144, 70–97 (1998)

    MathSciNet  MATH  Google Scholar 

  14. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  15. Dobrev, V., Kolev, T., Rieben, R.: High-order curvilinear finite element methods for Lagrangian hydrodynamics. SIAM J. Sci. Comput. 34, B606–B641 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Dukowicz, J.K., Meltz, B.: Vorticity errors in multidimensional lagrangian codes. J. Comput. Phys. 99, 115–134 (1992)

    MATH  Google Scholar 

  17. Gaburro, E., Boscheri, W., Chiocchetti, S., Klingenberg, C., Springel, V., Dumbser, M.: High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes. J. Comput. Phys. 407, 109167 (2020)

    MathSciNet  Google Scholar 

  18. Gaburro, E.: A unified framework for the solution of hyperbolic pde systems using high order direct Arbitrary-Lagrangian-Eulerian schemes on moving unstructured meshes with topology change. Arch. Comput. Methods Eng. (2020). https://doi.org/10.1007/s11831-020-09411-7

    Article  Google Scholar 

  19. Hirt, C.W., Armsden, A.A., Cook, J.L.: An arbitrary Lagrangian Eulerian computing method for all flow speed. J. Comput. Phys. 135, 203–216 (1997)

    MATH  Google Scholar 

  20. Hu, C., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Hui, W.H., Li, P.Y., Li, Z.W.: A unified coordinated system for solving the two-dimensional Euler equations. J. Comput. Phys. 153, 596–637 (1999)

    MathSciNet  MATH  Google Scholar 

  22. Jin, C.Q., Xu, K.: A unified moving grid gas-kinetic method in Eulerian space for viscous flow computation. J. Comput. Phys. 222, 155–175 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Kamm, J.R., Timmes, F.X.: On Efficient Generation of Numerically Robust Sedov Solutions, Technical Report LA-UR-07-2849, Los Alamos National Laboratory (2007)

  24. Kucharik, M., Shashkov, M.: Conservative multi-material remap for staggered multi-material Arbitrary Lagrangian-Eulerian methods. J. Comput. Phys. 258, 268–304 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic conservation laws. ESAIM Math. Model. Numer. Anal. 33, 547–571 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22, 656–672 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Li, J.Q., Du, Z.F.: A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers I. hyperbolic conservation laws. SIAM J. Sci. Comput. 38, 3046–3069 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Liu, X.D., Morgan, N.R., Burton, D.E.: A Lagrangian discontinuous Galerkin hydrodynamic method. Comput. Fluids 163, 68–85 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Maire, P.H., Abgrall, R., Breil, J., Ovadia, J.: A cell-centered Lagrangian scheme for two-dimensional compressible flow problems. SIAM J. Sci. Comput. 29, 1781–1824 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Maire, P.H.: A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry. J. Comput. Phys. 228, 6882–6915 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Maire, P.H., Nkonga, B.: Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics. J. Comput. Phys. 228, 799–821 (2009)

    MathSciNet  MATH  Google Scholar 

  32. Ni, G.X., Jiang, S., Xu, K.: Remapping-free ALE-type kinetic method for flow computations. J. Comput. Phys. 228, 3154–3171 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Noh, W.F.: Errors for calculations of strong shocks using artificial viscosity and an artificial heat flux. J. Comput. Phys. 72, 78–120 (1987)

    MATH  Google Scholar 

  34. Omang, M., Bøve, S., Trulsen, J.: SPH in spherical and cylindrical coordinates. J. Comput. Phys. 213, 391–412 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Persson, P.O., Bonet, J., Peraire, J.: Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains. Comput. Methods Appl. Mech. Eng. 198, 1585–1595 (2009)

    MATH  Google Scholar 

  36. Pan, L., Xu, K., Li, Q.B., Li, J.Q.: An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Navier-Stokes equations. J. Comput. Phys. 326, 197–221 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Pan, L., Xu, K.: High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions. Comput. Fluids 198, 104401 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Pan, L., Zhao, F.X., Xu, K.: High-order ALE gas-kinetic scheme with WENO reconstruction. J. Comput. Phys. 417, 109558 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Shi, J., Hu, C., Shu, C.W.: A Technique of Treating Negative Weights in WENO Schemes. J. Comput. Phys. 175, 108–127 (2002)

    MATH  Google Scholar 

  40. Tang, H.Z., Tang, T.: Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws. SIAM J. Numer. Anal. 41, 487–515 (2003)

    MathSciNet  MATH  Google Scholar 

  41. Thomas, P., Lombard, C.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17, 1030–1037 (1979)

    MathSciNet  MATH  Google Scholar 

  42. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, New York (2009)

    MATH  Google Scholar 

  43. Titarev, V.A., Toro, E.F.: ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17, 609–618 (2002)

    MathSciNet  MATH  Google Scholar 

  44. Titarev, V.A., Toro, E.F.: ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys. 204, 715–736 (2005)

    MathSciNet  MATH  Google Scholar 

  45. Xu, K.: Direct Modeling for Computational Fluid Dynamics: Construction and Application of Unfied Gas Kinetic Schemes. World Scientific (2015)

  46. 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, 289–335 (2001)

    MathSciNet  MATH  Google Scholar 

  47. Xu, X.H., Ni, G.X., Jiang, S.: A high-order moving mesh kinetic scheme based on WENO reconstruction for compressible flows on unstructured meshes. J. Sci. Comput. 57, 278–299 (2013)

    MathSciNet  MATH  Google Scholar 

  48. Zhao, F.X., Pan, L., Wang, S.H.: Weighted essentially non-oscillatory scheme on unstructured quadrilateral and triangular meshes for hyperbolic conservation laws. J. Comput. Phys. 374, 605–624 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Zhu, J., Qiu, J.X.: A new fifth order finite difference weno scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

    MathSciNet  MATH  Google Scholar 

  50. Zhu, J., Qiu, J.X.: New finite volume weighted essentially non-oscillatory scheme on triangular meshes. SIAM J. Sci. Comput. 40, 903–928 (2018)

    MathSciNet  Google Scholar 

  51. Zhu, J., Shu, C.W.: A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes. J. Comput. Phys. 406, 109212 (2020)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The current research of Liang Pan is supported by National Natural Science Foundation of China (11701038, 11702030) and the Fundamental Research Funds for the Central Universities (2018NTST19). The work of Kun Xu is supported by National Natural Science Foundation of China (11772281, 91852114), Hong Kong research grant council (16206617), and the National Numerical Windtunnel project.

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, China

    Liang Pan

  2. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Kun Xu

  3. Shenzhen Research Institute, Hong Kong University of Science and Technology, Shenzhen, China

    Kun Xu

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  1. Liang Pan
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Correspondence to Liang Pan.

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Pan, L., Xu, K. An Arbitrary-Lagrangian-Eulerian High-Order Gas-Kinetic Scheme for Three-Dimensional Computations. J Sci Comput 88, 8 (2021). https://doi.org/10.1007/s10915-021-01525-9

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