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A New Adaptation Strategy for Multi-resolution Method

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Abstract

Adaptive mesh refinement (AMR) and wavelet-based multi-resolution technique, which refine the spatial resolution in regions of interest and coarsen the mesh in other regions, are widely used in scientific computing for higher computational efficiency. The key component of AMR and the multi-resolution method is the adaptation strategy including the regularity estimation. In this paper, a new adaptation strategy for AMR and the multi-resolution method is proposed. Different from AMR, which measures the function regularity based on an empirical gradient operator, and the multi-resolution method with complicated wavelet analysis, the new approach examines the solution smoothness based on the high-order TENO reconstruction (Fu et al. in J Comput Phys 305:333–359, 2016). The core idea is that the TENO scheme not only provides the reconstructed data at the cell interface for flux evaluation but also classifies the local flow scales as smooth or nonsmooth on the discretized mesh. Since the scale-separation procedure is achieved in the spectral and characteristic space, the new adaptation strategy is weakly problem-dependent. Moreover, the extra complexity due to the empirical gradient computing or the wavelet analysis is eliminated. Based on Harten’s finite-volume multi-resolution method (Harten in J Comput Phys 115(2):319–338, 1994), a set of benchmark cases is simulated to validate the performance of the proposed method.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. 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)

    MathSciNet  MATH  Google Scholar 

  2. Fu, L., Hu, X.Y., Adams, N.A.: Single-step reinitialization and extending algorithms for level-set based multi-phase flow simulations. Comput. Phys. Commun. 221, 63–80 (2017)

    Article  MathSciNet  Google Scholar 

  3. Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53(3), 484–512 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82(1), 64–84 (1989)

    Article  MATH  Google Scholar 

  5. Domingues, M.O., Gomes, S.M., Roussel, O., Schneider, K.: Space-time adaptive multiresolution methods for hyperbolic conservation laws: applications to compressible Euler equations. Appl. Numer. Math. 59(9), 2303–2321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Reckinger, S.J., Livescu, D., Vasilyev, O.V.: Comprehensive numerical methodology for direct numerical simulations of compressible Rayleigh–Taylor instability. J. Comput. Phys. 313, 181–208 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hatori, T., Ito, A.M., Nunami, M., Usui, H., Miura, H.: Level-by-level artificial viscosity and visualization for MHD simulation with adaptive mesh refinement. J. Comput. Phys. 319, 231–241 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Donat, R., Martí, M.C., Martínez-Gavara, A., Mulet, P.: Well-balanced adaptive mesh refinement for shallow water flows. J. Comput. Phys. 257, 937–953 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berger, M.J., George, D.L., LeVeque, R.J., Mandli, K.T.: The GeoClaw software for depth-averaged flows with adaptive refinement. Adv. Water Resour. 34(9), 1195–1206 (2011)

    Article  Google Scholar 

  10. Liu, C., Hu, C.: Adaptive THINC-GFM for compressible multi-medium flows. J. Comput. Phys. 342, 43–65 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Han, L.H., Hu, X.Y., Adams, N.A.: Adaptive multi-resolution method for compressible multi-phase flows with sharp interface model and pyramid data structure. J. Comput. Phys. 262, 131–152 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Baeza, A., Mulet, P.: Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations. Int. J. Numer. Meth. Fluids 52(4), 455–471 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dumbser, M., Zanotti, O., Hidalgo, A., Balsara, D.S.: ADER-WENO finite volume schemes with space-time adaptive mesh refinement. J. Comput. Phys. 248, 257–286 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zanotti, O., Fambri, F., Dumbser, M., Hidalgo, A.: Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Comput. Fluids 118, 204–224 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Semplice, M., Coco, A., Russo, G.: Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction. J. Sci. Comput. 66(2), 692–724 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tang, Q., Chacon, L., Kolev, T.V., Shadid, J.N., Tang, X.-Z.: An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD. J. Comput. Phys. 454, 110967 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bell, J., Almgren, A., Beckner, V., Day, M., Lijewski, M., Nonaka, A., Zhang, W.: BoxLib user’s guide.https://github.com/BoxLib-Codes/BoxLib (2012)

  18. Hornung, R.D., Kohn, S.R.: Managing application complexity in the SAMRAI object-oriented framework. Concurr. Comput. Pract. Exp. 14(5), 347–368 (2002)

    Article  MATH  Google Scholar 

  19. Colella, P., Graves, D., Ligocki, T., Martin, D., Modiano, D., Serafini, D., Van Straalen, B.: Chombo software package for AMR applications design document (2000)

  20. MacNeice, P., Olson, K.M., Mobarry, C., De Fainchtein, R., Packer, C.: PARAMESH: a parallel adaptive mesh refinement community toolkit. Comput. Phys. Commun. 126(3), 330–354 (2000)

    Article  MATH  Google Scholar 

  21. de la Asunción, M., Castro, M.: Simulation of tsunamis generated by landslides using adaptive mesh refinement on GPU. J. Comput. Phys. 345, 91–110 (2017)

    Article  MathSciNet  Google Scholar 

  22. Fu, L., Hu, X.Y., Adams, N.A.: A physics-motivated Centroidal Voronoi Particle domain decomposition method. J. Comput. Phys. 335, 718–735 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Deiterding, R., Domingues, M.O., Gomes, S.M., Schneider, K.: Comparison of adaptive multiresolution and adaptive mesh refinement applied to simulations of the compressible Euler equations. SIAM J. Sci. Comput. 38(5), S173–S193 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vasilyev, O.V., Bowman, C.: Second-generation wavelet collocation method for the solution of partial differential equations. J. Comput. Phys. 165(2), 660–693 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Benkiewicz, K., Hayashi, K.: Two-dimensional numerical simulations of multi-headed detonations in oxygen-aluminum mixtures using an adaptive mesh refinement. Shock Waves 12(5), 385–402 (2003)

    Article  MATH  Google Scholar 

  26. Harten, A.: Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 115(2), 319–338 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Bihari, B.L., Harten, A.: Multiresolution schemes for the numerical solution of 2-D conservation laws I. SIAM J. Sci. Comput. 18(2), 315–354 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dumbser, M., Hidalgo, A., Zanotti, O.: High order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng. 268, 359–387 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fambri, F., Dumbser, M., Zanotti, O.: Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier–Stokes and resistive MHD equations. Comput. Phys. Commun. 220, 297–318 (2017)

    Article  MathSciNet  Google Scholar 

  30. 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  MathSciNet  MATH  Google Scholar 

  31. 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  MathSciNet  MATH  Google Scholar 

  32. 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  MathSciNet  MATH  Google Scholar 

  33. Fu, L.: A low-dissipation finite-volume method based on a new TENO shock-capturing scheme. Comput. Phys. Commun. 235, 25–39 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Fu, L.: A Hybrid Method with TENO Based Discontinuity Indicator for Hyperbolic Conservation Laws. Commun. Comput. Phys. 26, 973–1007 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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  MathSciNet  MATH  Google Scholar 

  37. 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(2), 61 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pirozzoli, S.: On the spectral properties of shock-capturing schemes. J. Comput. Phys. 219(2), 489–497 (2006)

    Article  MATH  Google Scholar 

  39. Fu, L., Hu, X.Y., Adams, N.A.: A family of high order targeted ENO schemes for compressible fluid simulations. In: The Ninth Symposium on Turbulence and Shear Flow Phenomena (TSFP-9), June 30–July 3 (2015)

  40. Haimovich, O., Frankel, S.H.: Numerical simulations of compressible multicomponent and multiphase flow using a high-order targeted ENO (TENO) finite-volume method. Comput. Fluids 146, 105–116 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sun, Z., Inaba, S., Xiao, F.: Boundary Variation Diminishing (BVD) reconstruction: a new approach to improve Godunov schemes. J. Comput. Phys. 322, 309–325 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  43. 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  MathSciNet  MATH  Google Scholar 

  44. Fu, L., Hu, X.Y., Adams, N.A.: Implicit large eddy simulations with a high-order TENO scheme. In: Tenth International Symposium on Turbulence and Shear Flow Phenomena (TSFP10), July 6–9, 2017 (2017)

  45. Fu, L., Hu, X.Y., Adams, N.A.: A new class of adaptive high-order TENO schemes for Hyperbolic Conservation Laws. In: ECCOMAS Thematic Conference: European Conference on High Order Nonlinear Numerical Methods for Evolutionary PDEs: Theory and Applications, March 27–31, 2017 (2017)

  46. 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(2), 311–345 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ren, Y.-X., Liu, M., Zhang, H.: A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 192(2), 365–386 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  48. Domingues, M.O., Gomes, S.M., Roussel, O., Schneider, K.: An adaptive multiresolution scheme with local time stepping for evolutionary PDEs. J. Comput. Phys. 227, 3758–3780 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  50. Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1), 25–34 (1994)

    Article  MATH  Google Scholar 

  51. Zhang, R., Zhang, M., Shu, C.-W.: On the order of accuracy and numerical performance of two classes of finite volume WENO schemes. Commun. Comput. Phys. 9(03), 807–827 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sun, Z., Teng, H., Xiao, F.: A slope constrained 4th order multi-moment finite volume method with weno limiter. Commun. Comput. Phys. 18(4), 901–930 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. 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  MathSciNet  MATH  Google Scholar 

  54. Gaburro, E., Dumbser, M.: A posteriori subcell finite volume limiter for general \( P_NP_M \) PNPM schemes: applications from gasdynamics to relativistic magnetohydrodynamics. J. Sci. Comput. 86(3), 1–41 (2021)

    Article  MATH  Google Scholar 

  55. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. Lax, P.D., Liu, X.-D.: Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19(2), 319–340 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  60. Kurganov, A., Tadmor, E.: Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Methods Partial Differ. Equ. 18(5), 584–608 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  61. Jung, C.-Y., Nguyen, T.B.: Fine structures for the solutions of the two-dimensional Riemann problems by high-order WENO schemes. Adv. Comput. Math. 44, 147–174 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  62. Jung, C.-Y., Nguyen, T.B.: Fine structures for the solutions of the two-dimensional Riemann problems by high-order WENO schemes. Adv. Comput. Math. 44(1), 147–174 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  63. Shen, Y., Zha, G.: Generalized finite compact difference scheme for shock/complex flowfield interaction. J. Comput. Phys. 230(12), 4419–4436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  64. Liska, R., Wendroff, B.: Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25(3), 995–1017 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  65. Vandenbroucke, B., De Rijcke, S.: The moving mesh code Shadowfax. Astron. Comput. 16, 109–130 (2016)

    Article  Google Scholar 

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Funding

Lin Fu acknowledges the fund from Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515011779), the fund from Shenzhen Municipal Central Government Guides Local Science and Technology Development Special Funds Funded Projects (No. 2021Szvup138), and the fund from Key Laboratory of Computational Aerodynamics, AVIC Aerodynamics Research Institute.

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  1. Lin Fu and Tian Liang contributed equally to this work.

Authors and Affiliations

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

    Lin Fu & Tian Liang

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

    Lin Fu

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

    Lin Fu

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Fu, L., Liang, T. A New Adaptation Strategy for Multi-resolution Method. J Sci Comput 93, 43 (2022). https://doi.org/10.1007/s10915-022-02012-5

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