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A Family of Fast Multi-resolution ENO Schemes for Compressible Flows

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Abstract

In this paper, a new class of high-order fast multi-resolution essentially non-oscillatory (FMRENO) schemes is proposed with an emphasis on both the performance and the computational efficiency. First, a new candidate stencil arrangement is developed for a multi-resolution representation of the local flow scales. A set of candidate stencils ranging from high- to low-order (from large to small stencils) is constructed in a hierarchical manner. Second, the monotonicity-preserving (MP) limiter is introduced as the regularity criterion of the candidate stencils. A candidate stencil, with which the reconstructed cell interface flux locates within the MP lower and upper bounds, is regarded to be smooth. Third, a multi-resolution stencil selection strategy, which prioritizes the stencils with better spectral property or higher-order accuracy meanwhile satisfying the MP criterion, is proposed. If all the candidate stencils are judged to be nonsmooth, the targeted stencil that violates the MP criterion the least is deployed as the final reconstruction instead. With this new framework, the desirable high-order accuracy is restored in the smooth regions while the sharp shock-capturing capability is achieved by selecting the targeted stencil satisfying the MP criterion most. Moreover, the new FMRENO schemes feature low numerical dissipation for resolving the broadband physical fluctuations by adaptively choosing the candidate stencil with better spectra or higher accuracy order based on the local flow regularity. Compared to the standard weighted/targeted essentially non-oscillatory (W/TENO) schemes, the computational efficiency is dramatically enhanced by avoiding the expensive evaluations of the classical smoothness indicators. A set of benchmark simulations demonstrate the performance of the new FMRENO schemes for handling complex fluid problems with a wide range of length scales.

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Data Availability

The data that support the findings of this study are available on request from the corresponding author, LF.

References

  1. Pirozzoli, S.: Numerical methods for high-speed flows. Annu. Rev. Fluid Mech. 43, 163–194 (2011)

    Article  MATH  Google Scholar 

  2. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)

    Article  MATH  Google Scholar 

  3. Fu, L., Karp, M., Bose, S.T., Moin, P., Urzay, J.: Shock-induced heating and transition to turbulence in a hypersonic boundary layer. J. Fluid Mech. 909, A8 (2021)

    Article  MATH  Google Scholar 

  4. Griffin, K.P., Fu, L., Moin, P.: Velocity transformation for compressible wall-bounded turbulent flows with and without heat transfer. Proc. Natl. Acad. Sci. 118, e2111144118 (2021)

    Article  Google Scholar 

  5. Von, N.J., Richtmyer, R.: A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21, 232 (1950)

    Article  MATH  Google Scholar 

  6. Jameson, A.: Analysis and design of numerical schemes for gas dynamics, 1: artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. Int. J. Comput. Fluid Dyn. 4, 171–218 (1994)

    Article  Google Scholar 

  7. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MATH  Google Scholar 

  10. Fu, L., Hu, X.Y., Adams, N.A.: A family of high-order targeted ENO schemes for compressible-fluid simulations. J. Comput. Phys. 305, 333–359 (2016)

    Article  MATH  Google Scholar 

  11. Antoniadis, A.F., Drikakis, D., Farmakis, P.S., Fu, L., Kokkinakis, I., Nogueira, X., Silva, P.A., Skote, M., Titarev, V., Tsoutsanis, P.: UCNS3D: an open-source high-order finite-volume unstructured CFD solver. Comput. Phys. Commun. 279, 108453 (2022)

    Article  Google Scholar 

  12. Tsoutsanis, P., Nogueira, X., Fu, L.: A short note on a 3D spectral analysis for turbulent flows on unstructured meshes. J. Comput. Phys. 111804 (2022)

  13. Henrick, A.K., Aslam, T., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)

    Article  MATH  Google Scholar 

  14. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)

    Article  MATH  Google Scholar 

  15. Don, W.-S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013)

    Article  MATH  Google Scholar 

  16. Hill, D.J., Pullin, D.I.: Hybrid tuned center-difference-WENO method for large Eddy simulations in the presence of strong shocks. J. Comput. Phys. 194, 435–450 (2004)

    Article  MATH  Google Scholar 

  17. Hu, X., Wang, Q., Adams, N.A.: An adaptive central-upwind weighted essentially non-oscillatory scheme. J. Comput. Phys. 229, 8952–8965 (2010)

    Article  MATH  Google Scholar 

  18. Acker, F., Borges, R.D.R., Costa, B.: An improved WENO-Z scheme. J. Comput. Phys. 313, 726–753 (2016)

    Article  MATH  Google Scholar 

  19. Suresh, A., Huynh, H.: Accurate monotonicity preserving schemes with Runge–Kutta time stepping. J. Comput. Phys. 136, 83–99 (1997)

    Article  MATH  Google Scholar 

  20. Zhang, X., Shu, C.-W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245–2258 (2012)

    Article  MATH  Google Scholar 

  21. Gerolymos, G., Sénéchal, D., Vallet, I.: Very-high-order WENO schemes. J. Comput. Phys. 228, 8481–8524 (2009)

    Article  MATH  Google Scholar 

  22. Zhu, J., Shu, C.-W.: A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J. Comput. Phys. 375, 659–683 (2018)

    Article  MATH  Google Scholar 

  23. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws. ESAIM Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique 33, 547–571 (1999)

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

    Article  MATH  Google Scholar 

  25. Fu, L., Hu, X.Y., Adams, N.A.: Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws. J. Comput. Phys. 349, 97–121 (2017)

    Article  MATH  Google Scholar 

  26. Fu, L., Hu, X.Y., Adams, N.A.: A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws. J. Comput. Phys. 374, 724–751 (2018)

    Article  MATH  Google Scholar 

  27. Fu, L., Hu, X.Y., Adams, N.A.: A targeted ENO scheme as implicit model for turbulent and genuine subgrid scales. Commun. Comput. Phys. 26, 311–345 (2019)

    Article  MATH  Google Scholar 

  28. Fu, L., Hu, X.Y., Adams, N.A.: Improved five- and six-point targeted essentially nonoscillatory schemes with adaptive dissipation. AIAA J. 57, 1143–1158 (2019)

    Article  Google Scholar 

  29. Fu, L.: A very-high-order TENO scheme for all-speed gas dynamics and turbulence. Comput. Phys. Commun. 244, 117–131 (2019)

    Article  Google Scholar 

  30. Fu, L.: A hybrid method with TENO based discontinuity indicator for hyperbolic conservation laws. Commun. Comput. Phys. 26, 973–1007 (2019)

    Article  MATH  Google Scholar 

  31. Li, Y., Fu, L., Adams, N.A.: A low-dissipation shock-capturing framework with flexible nonlinear dissipation control. J. Comput. Phys. 428, 109960 (2021)

    Article  MATH  Google Scholar 

  32. Takagi, S., Fu, L., Wakimura, H., Xiao, F.: A novel high-order low-dissipation TENO-THINC scheme for hyperbolic conservation laws. J. Comput. Phys. 452, 110899 (2022)

    Article  MATH  Google Scholar 

  33. Liang, T., Xiao, F., Shyy, W., Fu, L.: A fifth-order low-dissipation discontinuity-resolving TENO scheme for compressible flow simulation. J. Comput. Phys. 467, 111465 (2022)

    Article  MATH  Google Scholar 

  34. Ji, Z., Liang, T., Fu, L.: A class of new high-order finite-volume TENO schemes for hyperbolic conservation laws with unstructured meshes. J. Sci. Comput. 92, 61 (2022)

    Article  MATH  Google Scholar 

  35. Fu, L., Liang, T.: A new adaptation strategy for multi-resolution method. J. Sci. Comput. 93, 43 (2022)

    Article  MATH  Google Scholar 

  36. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)

    Article  MATH  Google Scholar 

  37. Wu, C., Wu, L., Zhang, S.: A smoothness indicator constant for sine functions. J. Comput. Phys. 419, 109661 (2020)

    Article  MATH  Google Scholar 

  38. He, Z., Zhang, Y., Gao, F., Li, X., Tian, B.: An improved accurate monotonicity-preserving scheme for the Euler equations. Comput. Fluids 140, 1–10 (2016)

    Article  MATH  Google Scholar 

  39. Fang, J., Li, Z., Lu, L.: An optimized low-dissipation monotonicity-preserving scheme for numerical simulations of high-speed turbulent flows. J. Sci. Comput. 56, 67–95 (2013)

    Article  MATH  Google Scholar 

  40. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)

    Article  MATH  Google Scholar 

  41. Xiao, F., Ii, S., Chen, C.: Revisit to the THINC scheme: a simple algebraic VOF algorithm. J. Comput. Phys. 230, 7086–7092 (2011)

    Article  MATH  Google Scholar 

  42. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)

    Article  MATH  Google Scholar 

  43. Rusanov, V.V.: Calculation of interaction of non-steady shock waves with obstacles. USSR J. Comput. Math. Phys. 267–279 (1961)

  44. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MATH  Google Scholar 

  45. Hu, X.Y., Wang, Q., Adams, N.A.: An adaptive central-upwind weighted essentially non-oscillatory scheme. J. Comput. Phys. 229, 8952–8965 (2010)

    Article  MATH  Google Scholar 

  46. Yamaleev, N.K., Carpenter, M.H.: A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228, 4248–4272 (2009)

    Article  MATH  Google Scholar 

  47. Zhu, J., Zhong, X., Shu, C.-W., Qiu, J.: Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes. J. Comput. Phys. 248, 200–220 (2013)

    Article  MATH  Google Scholar 

  48. Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954)

    Article  MATH  Google Scholar 

  49. Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  52. Xu, Z., Shu, C.W.: Anti-diffusive flux corrections for high order finite difference WENO schemes. J. Comput. Phys. 205, 458–485 (2005)

    Article  MATH  Google Scholar 

  53. Kurganov, A., Tadmor, E.: Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differ. Equ. 18, 584–608 (2002)

    Article  MATH  Google Scholar 

  54. Zeng, X., Scovazzi, G.: A frame-invariant vector limiter for flux corrected nodal remap in arbitrary Lagrangian–Eulerian flow computations. J. Comput. Phys. 270, 753–783 (2014)

    Article  MATH  Google Scholar 

  55. Zhang, S., Shu, C.-W.: A new smoothness indicator for the WENO schemes and its effect on the convergence to steady state solutions. J. Sci. Comput. 31, 273–305 (2007)

    Article  MATH  Google Scholar 

  56. Saad, M.A.: Compressible Fluid Flow. Englewood Cliffs (1985)

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Acknowledgements

The first author is partially supported by China Scholarship Council (NO. 201706290041). Lin Fu acknowledges the fund from the Research Grants Council (RGC) of the Government of Hong Kong Special Administrative Region (HKSAR) with RGC/ECS Project (No. 26200222), the fund from Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515011779), the fund from Key Laboratory of Computational Aerodynamics, AVIC Aerodynamics Research Institute, and the fund from the Project of Hetao Shenzhen-Hong Kong Science and Technology Innovation Cooperation Zone (No. HZQB-KCZYB-2020083).

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Authors and Affiliations

  1. Chair of Aerodynamics and Fluid Mechanics, School of Engineering and Design, Technical University of Munich, 85748, Garching, Germany

    Yue Li & Nikolaus A. Adams

  2. Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Lin Fu

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

    Lin Fu

  4. HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute, Futian, Shenzhen, China

    Lin Fu

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

    Lin Fu

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  1. Yue Li
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Li, Y., Fu, L. & Adams, N.A. A Family of Fast Multi-resolution ENO Schemes for Compressible Flows. J Sci Comput 94, 44 (2023). https://doi.org/10.1007/s10915-022-02095-0

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