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High Order Finite Volume Schemes for Solving the Non-Conservative Convection Equations on the Unstructured Grids

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Abstract

In this paper, a high order finite volume scheme for solving the non-conservative convection equations on the unstructured grids is proposed. It is found that when the non-conservative convection equations are rewritten into the conservative form with additional source term, the direct application of the finite volume scheme using high order reconstruction will produce numerical instability. To solve this problem, we propose in the present paper to solve the integral form of the non-conservative convection equations. To account for the upwinding effect, a convective reconstruction technique is proposed. The proposed method is applied to solve a linear advection equation and the eikonal equation in time dependent non-conservative form. An artificial viscosity term is added to handle the singularity of the equation. The numerical results show that the proposed numerical scheme can achieve high order accuracy and is very robust.

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References

  1. Wang, Q., Ren, Y.X., Li, W.: Compact high order finite volume method on unstructured grids I: basic formulations and one-dimensional schemes. J. Comput. Phys. 314, 863–882 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Wang, Q., Ren, Y.X., Li, W.: Compact high order finite volume method on unstructured grids II: extension to two-dimensional Euler equations. J. Comput. Phys. 314, 883–908 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wang, Q., Ren, Y.X., Pan, J., et al.: Compact high order finite volume method on unstructured grids III: variational reconstruction[J]. J. Comput. Phys. 337, 1–26 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhang, Y.S., Ren, Y.X., Wang, Q.: Compact high order finite volume method on unstructured grids IV: explicit multi-step reconstruction schemes on compact stencil[J]. J. Comput. Phys. 396, 161–192 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pan, J., Ren, Y.X., Sun, Y.: High order sub-cell finite volume schemes for solving hyperbolic conservation laws II: extension to two-dimensional systems on unstructured grids[J]. J. Comput. Phys. 338, 165–198 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138(2), 251–285 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J Comput Phys 131(2), 267–279 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Shu, C.W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Romero, J., Asthana, K., Jameson, A.: A simplified formulation of the flux reconstruction method[M]. Plenum Press, New York (2016)

    Book  MATH  Google Scholar 

  10. Huynh HT. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: 18th AIAA Computational Fluid Dynamics Conference p. 4079 (2007)

  11. Wang, Z.J., Gao, H.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids[J]. J. Comput. Phys. 228(21), 8161–8186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dumbser, M., Balsara, D.S., Toro, E.F., et al.: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes[J]. J. Comput. Phys. 227(18), 8209–8253 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, L., Wei, L., He, L.X., et al.: A class of hybrid DG/FV methods for conservation laws I: basic formulation and one-dimensional systems[J]. J. Comput. Phys. 231(4), 1081–1103 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, L., Wei, L., He, L.X., et al.: A class of hybrid DG/FV methods for conservation laws II: two-dimensional cases[J]. J. Comput. Phys. 231(4), 1104–1120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ii, S., Xiao, F.: CIP/multi-moment finite volume method for Euler equations: a semi-Lagrangian characteristic formulation[J]. J. Comput. Phys. 222(2), 849–871 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, C., Xiao, F.: Shallow water model on cubed-sphere by multi-moment finite volume method[J]. J. Comput. Phys. 227(10), 5019–5044 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Xie, B., Deng, X., Sun, Z., et al.: A hybrid pressure–density-based Mach uniform algorithm for 2D Euler equations on unstructured grids by using multi-moment finite volume method[J]. J. Comput. Phys. 335, 637–663 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pan, L., Xu, K.: A compact third-order gas-kinetic scheme for compressible Euler and Navier-Stokes equations[J]. Commun. Comput. Phys. 18(04), 985–1011 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pan, L., Xu, K.: A third-order compact gas-kinetic scheme on unstructured meshes for compressible Navier-Stokes solutions[J]. J. Comput. Phys. 318, 327–348 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li L, Liu X, Lou J, Luo H, Nishikawa H, and Ren Y. “A discontinuous Galerkin method based on variational reconstruction for compressible flows on arbitrary grids”. 56th AIAA Aerospace Sciences Meeting, AIAA Paper p. 0831, Kissimmee, Florida, (2018)

  21. Li L, Liu X, Lou J, Luo H, Nishikawa H, and Ren Y. “A Finite Volume Method Based on Variational Reconstruction for Compressible Flows on Arbitrary Grids”. 23rd AIAA Computational Fluid Dynamics Conference, AIAA Paper 2017–3097, Denver, Colorado, (2017)

  22. Nishikawa H, An implicit gradient method for cell-centered finite-volume solver on unstructured grids. In: AIAA Scitech 2019 Forum, AIAA p. 1155 (2019)

  23. Cheng, J., Liu, T.: A variational reconstructed discontinuous Galerkin method for the steady-state compressible flows on unstructured grids[J]. J. Comput. Phys. 380, 65–87 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, X., Xia, Y., Luo, H.: A reconstructed discontinuous Galerkin method for compressible turbulent flows on 3D curved grids[J]. Comput. Fluids 160, 26–41 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sethian, J.A.: Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science[M]. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  26. Gremaud, P.A., Kuster, C.M.: Computational study of fast methods for the eikonal equation[J]. SIAM J. Sci. Comput. 27(6), 1803–1816 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jiang, G.S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations[J]. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes[J]. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shu, C.W., Osher, S.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations[J]. SIAM J. Numer. Anal. 28(4), 907–922 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, Y.T., Zhao, H.K., Qian, J.: High order fast sweeping methods for static Hamilton-Jacobi equations[J]. J. Sci. Comput. 29(1), 25–56 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, F., Shu, C.W., Zhang, Y.T., et al.: A second order discontinuous Galerkin fast sweeping method for eikonal equations[J]. J. Comput. Phys. 227(17), 8191–8208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, Y.T., Chen, S., Li, F., et al.: Uniformly accurate discontinuous Galerkin fast sweeping methods for eikonal equations[J]. SIAM J. Sci. Comput. 33(4), 1873–1896 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cheng, Y., Wang, Z.: A new discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations[J]. J. Comput. Phys. 268, 134–153 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Luo, S.: A uniformly second order fast sweeping method for eikonal equations[J]. J. Comput. Phys. 241, 104–117 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhao, H.: Parallel implementations of the fast sweeping method[J]. J. Comput. Math. 25(4), 421–429 (2007)

    MathSciNet  Google Scholar 

  36. Detrixhe, M., Gibou, F., Min, C.: A parallel fast sweeping method for the eikonal equation[J]. J. Comput. Phys. 237(1), 46–55 (2013)

    Article  MathSciNet  Google Scholar 

  37. Yang, J., Stern, F.: A highly scalable massively parallel fast marching method for the eikonal equation[J]. J. Comput. Phys. 332, 333–362 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Fares, E., SchröDer, W.: A differential equation for approximate wall distance[J]. Int. J. Numer. Meth. Fluids 39(8), 743–762 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhou C, Wang Z J. 53rd AIAA Aerospace Sciences Meeting - CPR High-order Discretization of the RANS Equations with the SA Model[C]. In: AIAA Aerospace Sciences Meeting, (2015)

  40. Schoenawa, S., Hartmann, R.: Discontinuous Galerkin discretization of the Reynolds-averaged Navier-Stokes equations with the shear-stress transport model[J]. J. Comput. Phys. 262, 194–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xia, H., Tucker, P.G.: Finite volume distance field and its application to medial axis transforms[J]. Int. J. Numer. Meth. Eng. 82(1), 114–134 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Castro, M., Gallardo, J.M., Parés, C.: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems[J]. Math. Comput. 75(255), 1103–1134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework[J]. SIAM J. Numer. Anal. 44(1), 300–321 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Castro, M., Pardo, A., Parés, C., Toro, E.F.: On some fast well-balanced first order solvers for nonconservative systems[J]. Math. Comput. 79(271), 1427–1472 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Dumbser, M., Toro, E.F.: A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems[J]. J. Sci. Comput. 48(1–3), 70–88 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Dumbser, M., Balsara, D.S.: A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems[J]. J. Comput. Phys. 304, 275–319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products[J]. J. De Mathématiques Pures Et Appliquées 74(6), 483–548 (1995)

    MathSciNet  MATH  Google Scholar 

  48. Xu, J.L., Yan, C., Fan, J.J.: Computations of wall distances by solving a transport equation[J]. Appl. Math. Mech. 32(2), 141–150 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  49. Luo, H., Baum, J.D., Löhner, R.: A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids[J]. J. Comput. Phys. 227(20), 8875–8893 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Li, W.A., Ren, Y.X.: High-order k-exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids[J]. Int. J. Numer. Methods Fluids 70(6), 742–763 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, Q.J., Ren, Y.X., Sun, Z.S., Sun, Y.T.: Low dispersion finite volume scheme based on reconstruction with minimized dispersion and controllable dissipation. Sci. China Phys. Mech. Astron. 56(2), 423–431 (2013)

    Article  Google Scholar 

  53. Wang, Q.J., Ren, Y.X.: An accurate and robust finite volume scheme based on the spline interpolation for solving the Euler and Navier-Stokes equations on non-uniform curvilinear grids[J]. J. Comput. Phys. 284, 648–667 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids[J]. J. Comput. Phys. 31(3), 335–362 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  55. LeVeque, R.J.: High-resolution conservative algorithms for advection in incompressible flow[J]. Siam J. Numer. Anal. 33(2), 627–665 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the China Postdoctoral Science Foundation under Grant No. 2019M660613, the national numerical wind tunnel project, and Project 91752114 of NSFC.

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Authors and Affiliations

  1. Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

    Qian-Min Huang & Yu-Xin Ren

  2. Mathematics Section, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    Qian Wang

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  1. Qian-Min Huang
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Contributions

All authors contributed to the study conception and design based on the initial idea of YR. The numerical schemes, coding and data processing were carried out by QH based on the code developed by QW. The first draft of the manuscript was written by QH and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yu-Xin Ren.

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Huang, QM., Ren, YX. & Wang, Q. High Order Finite Volume Schemes for Solving the Non-Conservative Convection Equations on the Unstructured Grids. J Sci Comput 88, 37 (2021). https://doi.org/10.1007/s10915-021-01538-4

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