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Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

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Abstract

This paper presents a second-order accurate adaptive Godunov method for two-dimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41:487–515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometry-based conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.

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References

  1. Abgrall, R.: How to prevent oscillations in multicomponent flow calculations: A quasi conservative approach. J. Comput. Phys. 125, 150–160 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abgrall, R., Karni, S.: Computations of compressible multifluids. J. Comput. Phys. 169, 594–623 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abgrall, R., Saurel, R.: Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186, 361–396 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barth, T.J., Jespersen, D.C.: The design and application of upwind schemes on unstructured meshes. AIAA Paper No. 89-0366 (1989)

  5. Beckett, G., Mackenzie, J.A., Robertson, M.L.: An r-adaptive finite element method for the solution of the two-dimensional phase-field equations. Commun. Comput. Phys. 1, 805–826 (2006)

    Google Scholar 

  6. Brackbill, J.U.: An adaptive grid with directional control. J. Comput. Phys. 108, 38–50 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cao, W.M., Huang, W.Z., Russell, R.D.: An r-adaptive finite element method based upon moving mesh PDEs. J. Comput. Phys. 149, 221–244 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ceniceros, H.D., Hou, T.Y.: An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 172, 609–639 (2001)

    Article  MATH  Google Scholar 

  9. Chertock, A., Kurganov, A.: Conservative locally moving mesh method for multifluid flows. In: Proc. 4th Inter. Sym. on Finite Volumes for Complex Appl., Marrakech, pp. 273–284 (2005)

  10. Davis, S.F., Flaherty, J.E.: An adaptive finite element method for initial-boundary value problems for partial differential equations. SIAM J. Sci. Stat. Comput. 3, 6–27 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  11. Di, Y.N., Li, R., Tang, T., Zhang, P.W.: Moving mesh finite element methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 26, 1036–1056 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dvinsky, A.S.: Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comput. Phys. 95, 450–476 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Emery, A.F.: An evaluation of several differencing methods for inviscid fluid flow problem. J. Comput. Phys. 2, 306–331 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  14. Godunov, S.K.: Finite difference method for numerical computation of discontinuous solution of the equations of fluid dynamics. Mat. Sb. 47, 271–306 (1959)

    MathSciNet  Google Scholar 

  15. Han, J.Q., Tang, H.Z.: An adaptive moving mesh method for multidimensional ideal magnetohydrodynamics. J. Comput. Phys. 220, 791–812 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hou, T.Y., LeFloch, P.G.: Why nonconservative schemes converge to wrong solutions: Error analysis. Math. Comput. 62, 497–530 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Huang, W.Z.: Metric tensors for anisotropic mesh generation. J. Comput. Phys. 204, 633–665 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Huang, W.Z.: Mathematical principles of anisotropic mesh adaptation. Commun. Comput. Phys. 1, 276–310 (2006)

    Google Scholar 

  20. Jameson, A., Mavriplis, D.: Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh. AIAA Paper No. 85-0435 (1985)

  21. Jia, P.Y., Jiang, S., Zhao, G.P.: Two-dimensional compressible multimaterial flow calculations in a unified coordinate system. Comput. Fluids 35, 168–188 (2006)

    Article  MATH  Google Scholar 

  22. Jiang, S., Ni, G.X.: A γ-model BGK scheme for compressible multifluids. Int. J. Numer. Meth. Fluids 46, 163–182 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Karni, S.: Hybrid multifluid algorithms. SIAM J. Sci. Comput. 17, 1019–1039 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Karni, S.: Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112, 31–43 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Larouturou, B.: How to preserve the mass fraction positive when computing compressible multi-component flows. J. Comput. Phys. 95, 59–84 (1991)

    Article  MathSciNet  Google Scholar 

  26. Li, R., Tang, T., Zhang, P.W.: Moving mesh methods in multiple dimensions based on harmonic maps. J. Comput. Phys. 170, 562–588 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Li, R., Tang, T., Zhang, P.W.: A moving mesh finite element algorithm for singular problems in two and three space dimensions. J. Comput. Phys. 177, 365–393 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lian, Y.S., Xu, K.: A gas-kinetic schemes for multimaterial flows and its application in chemical reactions. J. Comput. Phys. 163, 349–375 (2000)

    Article  MATH  Google Scholar 

  29. Liu, T.G., Khoo, B.C., Yeo, K.S.: The simulation of compressible multi-medium flow. Part I: A new methodology with applications to 1D gas-gas and gas-water cases. Comput. Fluids 30, 291–314 (2001)

    Article  MATH  Google Scholar 

  30. Miller, K., Miller, R.N.: Moving finite element I. SIAM J. Numer. Anal. 18, 1019–1032 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  31. Niceno, B.: A two-dimensional quality mesh generator. http://www-dinma.univ.trieste.it/nirftc/research/easymesh/

  32. Ren, W.Q., Wang, X.P.: An iterative grid redistribution method for singular problems in multiple dimensions. J. Comput. Phys. 159, 246–273 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  33. Saurel, R., Abgrall, R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  34. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  35. Shyue, K.M.: A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions. J. Comput. Phys. 215, 219–244 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Tan, Z.J., Zhang, Z.R., Tang, T., Huang, Y.Q.: Moving mesh methods with locally varying time steps. J. Comput. Phys. 200, 347–367 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Tang, H.Z.: On the sonic point glitch. J. Comput. Phys. 202, 507–532 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Tang, H.Z.: A moving mesh method for the Euler flow calculations using a directional monitor function. Commun. Comput. Phys. 1, 656–676 (2006)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  40. Tang, H.Z., Tang, T., Zhang, P.W.: An adaptive mesh redistribution method for nonlinear Hamilton-Jacobi equations in two- and three-dimensions. J. Comput. Phys. 188, 543–572 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  41. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2nd edn. Springer, Berlin (1999)

    MATH  Google Scholar 

  42. van Leer, B.: Towards the ultimate conservative difference schemes V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)

    Article  Google Scholar 

  43. Wang, J., Liu, R.X.: A comparative study of finite volume methods on unstructured meshes for simulation of 2D shallow water wave problems. Math. Comput. Simul. 53, 171–184 (2000)

    Article  Google Scholar 

  44. Winslow, A.: Numerical solution of the quasi-linear Poisson equation. J. Comput. Phys. 1, 149–172 (1967)

    Article  MathSciNet  Google Scholar 

  45. Woodward, P.R., Colella, P.: The numerical simulation of two dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  46. Xu, K.: BGK-based scheme for multicomponent flow calculations. J. Comput. Phys. 134, 122–133 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  47. Zhang, Z.R.: Moving mesh method with conservative interpolation based on L 2-projection. Commun. Comput. Phys. 1, 930–944 (2006)

    Google Scholar 

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Authors and Affiliations

  1. LMAM and CCSE, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Guoxian Chen, Huazhong Tang & Pingwen Zhang

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  1. Guoxian Chen
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  2. Huazhong Tang
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  3. Pingwen Zhang
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Correspondence to Huazhong Tang.

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Chen, G., Tang, H. & Zhang, P. Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes. J Sci Comput 34, 64–86 (2008). https://doi.org/10.1007/s10915-007-9162-8

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  • DOI: https://doi.org/10.1007/s10915-007-9162-8

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