Advertisement

High Order Compact Generalized Finite Difference Methods for Solving Inviscid Compressible Flows

  • Xue-Li Li
  • Yu-Xin RenEmail author
Article

Abstract

This paper presents a novel generalized finite difference method that can achieve arbitrary order of accuracy on a compact stencil nodal set. Accurate reconstruction and flux evaluation are two key steps to achieve high order spatial accuracy. A newly developed variational reconstruction approach is utilized to obtain the piecewise higher order polynomial distribution of flow variables. The implementation of boundary conditions is of critical importance and a flexible variational extrapolation technique is proposed for the high order boundary treatment. The numerical flux derivatives are evaluated using a simple and efficient hybrid approach, in which the linear and high order terms of the flux function are treated differently. Several test cases are solved to verify the accuracy, efficiency, and shock capturing capability of the proposed numerical schemes for inviscid compressible flows.

Keywords

Compact schemes High order boundary treatment Hybrid flux evaluation approach Variational reconstruction 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11672160 and 91752114) and national numerical wind tunnel project under contract number 2018-ZT4A07.

References

  1. 1.
    Katz, A., Jamenson, A.: A comparison of various meshless schemes within a unified algorithm. AIAA Paper 2009-596 (2009)Google Scholar
  2. 2.
    Katz, A., Jameson, A.: Meshless scheme based on alignment constraints. AIAA J. 48(11), 2501–2511 (2010)Google Scholar
  3. 3.
    Hashemi, Y., Jahangirian, A.: Implicit fully mesh-less method for compressible viscous flow calculations. J. Comput. Appl. Math. 235, 4687–4700 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Su, X.R., Yamamoto, S., Nakahashi, K.: Analysis of a meshless solver for high Reynolds number flow. J. Comput. Phys. 72, 505–527 (2013)MathSciNetGoogle Scholar
  5. 5.
    Sundar, D.S., Yeo, K.S.: A high order meshless method with compact support. J. Comput. Phys. 272, 70–87 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ding, H., Shu, C., Yeo, K.S.: Development of least-square-based two-dimensional finite-difference schemes and their application to simulate natural convection in a cavity. Comput. Fluids 33, 137–154 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Tota, P.V., Wang, Z.J.: Meshfree Euler solver using local radial basis functions for inviscid compressible flows. AIAA Paper 2007-4581 (2007)Google Scholar
  8. 8.
    Jaisankar, S., Shivashankar, K., Raghurama Rao, S.V.: A grid-free central scheme for inviscid compressible flows. AIAA Paper 2007-3946 (2007)Google Scholar
  9. 9.
    Anandhanarayanan, K., Krishnamurthy, R., Debasis, C.: Development and validation of a grid-free viscous solver. AIAA J. 54(10), 3310–3313 (2016)Google Scholar
  10. 10.
    Batina, J.T.: A gridless Euler/Navier–Stokes solution algorithm for complex-aircraft applications. AIAA Paper 93-0333 (1993)Google Scholar
  11. 11.
    Morinishi, K.: Gridless type solution for high Reynolds number multielement flow fields. AIAA Paper 95-1856 (1995)Google Scholar
  12. 12.
    Liu, J.L., Su, S.J.: A potentially gridless solution method for the compressible Euler/Navier-Stokes equation. AIAA Paper 96-0526 (1996)Google Scholar
  13. 13.
    Kirshman, D.J., Liu, F.: Gridless boundary condition treatment for a non-body-conforming mesh. AIAA Paper 2002-3285 (2002)Google Scholar
  14. 14.
    Koh, E.P.C., Tsai, H.M.: Euler solution using cartesian grid with a gridless least-squares boundary treatment. AIAA J. 43(2), 246–255 (2005)Google Scholar
  15. 15.
    Luo, H., Baum, J.D., Lӧhner, R.: A hybrid building-block and gridless method for compressible flows. AIAA Paper 2006-3710 (2006)Google Scholar
  16. 16.
    Ma, Z.H., Chen, H.Q., Zhou, C.H.: A study of point moving adaptivity in gridless method. Comput. Methods Appl. Mech. Eng. 197, 1926–1937 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sridar, D., Balakrishnan, N.: An upwind finite difference scheme for meshless solvers. J. Comput. Phys. 189, 1–29 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Munikrishna, N., Balakrishnan, N.: Turbulent flow computations on a hybrid Cartesian point distribution using meshless solver LSFD-U. Comput. Fluids 40(1), 118–138 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ortega, E., Oñate, E., Idelsohn, S.: A finite point method for adaptive three-dimensional compressible flow calculations. Int. J. Numer. Meth. Fluids 60, 937–971 (2009)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lӧhner, R., Sacco, C., Onate, E., Idelsohn, S.: A finite point method for compressible flow. Int. J. Numer. Methods Eng. 53, 1765–1779 (2002)zbMATHGoogle Scholar
  21. 21.
    Ortega, E., Oñate, E., Idelsohn, S., Flores, F.: Application of the finite point method to high-Reynolds number compressible flow problems. Int. J. Numer. Methods Fluids 74, 732–748 (2014)MathSciNetGoogle Scholar
  22. 22.
    Chung, K.C.: A generalized finite-difference method for heat transfer problems of irregular geometries. Numer. Heat Transf. 4, 345–357 (1981)Google Scholar
  23. 23.
    Morinishi, K.: Gridless type-generalized finite difference method. In: Computational Fluid Dynamics for the 21st Century: Notes on Numerical Fluid Mechanics vol. 78, pp. 43–58 (2001)Google Scholar
  24. 24.
    Shu, C., Ding, H., Chen, H.Q., Wang, T.G.: An upwind local RBF-DQ method for simulation of inviscid compressible flows. Comput. Methods Appl. Mech. Eng. 194, 2001–2017 (2005)zbMATHGoogle Scholar
  25. 25.
    Borthakur, M.P., Biswas, A.: A novel Hermite Taylor least square based meshfree framework with adaptive upwind scheme for two dimensional incompressible flows. Comput. Fluids 130, 37–48 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Traska, N., Maxeya, M., Hu, X.: Compact moving least squares: an optimization framework for generating high-order compact meshless discretizations. J. Comput. Phys. 326, 596–611 (2016)MathSciNetGoogle Scholar
  27. 27.
    Traska, N., Maxeya, M., Hu, X.: A compatible high-order meshless method for the Stokes equations with applications to suspension flows. J. Comput. Phys. 355, 310–326 (2018)MathSciNetGoogle Scholar
  28. 28.
    Weinan, E., Liu, J.G.: Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126(1), 122–138 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Li, X.L., Ren, Y.X., Li, W.: Construction of the high order accurate generalized finite difference schemes for inviscid compressible flows. Commun. Comput. Phys. 25(2), 481–507 (2019)MathSciNetGoogle Scholar
  31. 31.
    Abgrall, R., Larat, A., Ricchiuto, M.: Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes. J. Comput. Phys. 230(11), 4103–4136 (2011)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Wang, Q., Ren, Y.X., Li, W.: Compact high order finite volume method on unstructured grids III: variational reconstruction. J. Comput. Phys. 337, 1–26 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Gosh, A.K., Deshpande, S.M.: Least squares kinetic upwind method for inviscid compressible flows. AIAA Paper 95-36586 (1995)Google Scholar
  34. 34.
    Deshpande, S.M., Ramesh, V., Malagi, K., et al.: Least squares kinetic mesh-free method. Def. Sci. J. 60(6), 583–597 (2010)Google Scholar
  35. 35.
    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)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Sun, Z.S., Ren, Y.X., Zha, B., et al.: High order boundary conditions for high order finite difference schemes on curvilinear coordinates solving compressible flows. J. Sci. Comput. 65, 790–820 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Li, W., Ren, Y.X.: The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids II: extension to high order finite volume schemes. J. Comput. Phys. 231, 4053–4077 (2012)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 130, 202–228 (1996)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Hu, C.Q., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Li, W., Ren, Y.X.: High-order k-exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids. Int. J. Numer. Methods Fluids 70(6), 742–763 (2012)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Toro, E.F., Titarev, V.A.: Derivative Riemann solvers for systems of conservation laws and ADER methods. J. Comput. Phys. 212, 150–165 (2006)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Roe, P.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Wang, Z.J., Gao, H.Y.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume-difference methods for conservation laws on mixed grids. J. Comput. Phys. 228, 8161–8186 (2009)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge–Kutta methods. Appl. Numer. Math. 58(11), 1675–1686 (2008)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Arnone, A., Liou, M.S., Povinelli, L.A.: Integration of Navier-Stokes equations using dual time stepping and a multigrid method. AIAA J. 33(6), 985–990 (1995)zbMATHGoogle Scholar
  46. 46.
    Zhang, L.P., Wang, Z.J.: A block LU-SGS implicit dual time-stepping algorithm for hybrid dynamic meshes. Comput. Fluids 33(7), 891–916 (2004)zbMATHGoogle Scholar
  47. 47.
    Avesani, D., Dumbser, M., Bellin, A.: A new class of moving-least-squares WENO–SPH schemes. J. Comput. Phys. 270, 278–299 (2014)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Krivodonova, L., Berger, M.: High order accurate implementation of solid wall boundary conditions in curved geometries. J. Comput. Phys. 211, 492–512 (2006)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Wang, Z.J., Sun, Y.: Curvature-based wall boundary condition for the Euler equations on unstructured grids. AIAA J. 41(1), 27–33 (2003)MathSciNetGoogle Scholar
  50. 50.
    Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83(1), 32–78 (1989)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Kirshman, D.J., Liu, F.: A gridless boundary condition method for the solution of the Euler equtaions on embedded Cartesian meshes with multigrid. J. Comput. Phys. 201, 119–147 (2004)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.School of Aerospace EngineeringTsinghua UniversityBeijingChina

Personalised recommendations