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A Time-Accurate Dynamic Mesh Approach for High-Order Shock Computation

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Abstract

A dynamic mesh approach for conservation laws has been developed to compute discontinuities with high accuracy. The basic idea is to move the mesh interface at the speed of discontinuity at that interface. For any 1D scalar conservation law with piece-wise initial data, such a scheme is exact and unconditionally stable. Unfortunately, the exactness property for piecewise constant initial data does not carry over to non-linear systems. Instead, we couple this idea with a high-order discontinuous method to compute shocks with high-order accuracy. We only move mesh interfaces that are aligned with a shock or contact while other mesh interfaces remain stationary. This dynamic mesh approach is tested with several benchmark problems ranging from scalar conservation laws to 1D Euler equations. For a steady transonic flow through a convergent-divergent nozzle, super-convergence of order 2P or higher has been observed for the location of the shock with P being the degree of solution polynomial while the L1 solution error in both the shock-upstream and shock-downstream regions appears to be order P + 1. For unsteady test cases, the fitted shock at all orders of accuracy agrees very well with the exact or reference solution. Both strong and weak shocks are automatically detected and fitted in the Shu-Osher shock-entropy interaction problem, and no stabilizing techniques such as solution limiting or filtering are needed. These test cases have demonstrated the potential of the present approach in multiple dimensions.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Bonfiglioli, A., Paciorri, R.: Convergence analysis of shock-capturing and shock-fitting solutions on unstructured grids. AIAA J. 52(7), 1404–1416 (2014). https://doi.org/10.2514/1.J052567

    Article  MATH  Google Scholar 

  2. Carpenter, M.H., Casper, J.H.: Accuracy of shock capturing in two spatial dimensions. AIAA J. 37(9), 1072–1079 (1999). https://doi.org/10.2514/2.835

    Article  MATH  Google Scholar 

  3. Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989). https://doi.org/10.1090/S0025-5718-1989-0983311-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Corrigan, A., Kercher, A.D., Kessler, D.A.: A moving discontinuous Galerkin finite element method for flows with interfaces. Int. J. Numer. Meth. Fluids 89, 362–406 (2019). https://doi.org/10.1002/fld.4697

    Article  MathSciNet  MATH  Google Scholar 

  5. Dahl, M.D. Ed.: Third computational aeroacoustics (CAA) workshop on benchmark problems, technical report NASA/CP-2000-209790, NASA (2000).

  6. D’Aquila, L.M., Helenbrook, B.T., Mazaheri, A.: A novel stabilization method for high-order shock fitting with finite element methods. J. Comput. Phys. 430, 110096 (2021). https://doi.org/10.1016/j.jcp.2020.110096

    Article  MathSciNet  MATH  Google Scholar 

  7. Gnoffo, P., White, J.A.: Computational aerothermodynamic simulation issues on unstructured grids. AIAA-2004–2371; 2004 https://doi.org/10.2514/6.2004-2371

  8. Hughes, T.J.R., Liu, W.K., Zimmermann, T.K.: Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29(3), 329–349 (1981). https://doi.org/10.1016/0045-7825(81)90049-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper 4079 (2007). https://doi.org/10.2514/6.2007-4079

  10. Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202 (1996). https://doi.org/10.1006/jcph.1996.0130

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, Y., Vinokur, M., Wang, Z.J.: Discontinuous spectral difference method for conservation laws on unstructured grids. J. Comput. Phys. 216, 780–801 (2006). https://doi.org/10.1016/j.jcp.2006.01.024

    Article  MathSciNet  MATH  Google Scholar 

  12. Luo, H., Absillis, G., Nourgaliev, R.: A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows. J. Comput. Phys. 445, 110618 (2021). https://doi.org/10.1016/j.jcp.2021.110618

    Article  MathSciNet  MATH  Google Scholar 

  13. Moretti, G.: Computation of flows with shocks. Annu. Rev. Fluid Mech. 19(1), 313–337 (1987). https://doi.org/10.1146/annurev.fl.19.010187.001525

    Article  MATH  Google Scholar 

  14. Moretti, G.: A techniques for integrating two-dimensional Euler equations. Comput. Fluids 15(1), 59–75 (1987). https://doi.org/10.1016/0045-7930(87)90005-3

    Article  MathSciNet  MATH  Google Scholar 

  15. Naudet, C.J., Zahr, M.J.: A space-time high-order implicit shock tracking method for shock-dominated unsteady flows. J. Comput. Phys. 501, 112792 (2024). https://doi.org/10.1016/j.jcp.2024.112792

    Article  MathSciNet  MATH  Google Scholar 

  16. Nishkawa, H.: Uses of zero and negative volume elements for node-centered edge-based discretization, AIAA-2017-4295. https://doi.org/10.2514/6.2017-4295

  17. Prakash, A., Parsons, N., Wang, X., Zhong, X.: High-order shock-fitting methods for direct numerical simulation of hypersonic flow with chemical and thermal nonequilibrium. J. Comput. Phys. 230(23), 8474–8507 (2011). https://doi.org/10.1016/j.jcp.2011.08.001

    Article  MathSciNet  MATH  Google Scholar 

  18. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)

  19. Richtmyer, R.D.: A proposed method for the calculation of shocks. Technical Report LA–671, Los Alamos Scientific Laboratory (1948)

  20. Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981). https://doi.org/10.1016/0021-9991(81)90128-5

    Article  MathSciNet  MATH  Google Scholar 

  21. Shu, C., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988). https://doi.org/10.1016/0021-9991(88)90177-5

    Article  MathSciNet  MATH  Google Scholar 

  22. Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27(1), 1–31 (1978). https://doi.org/10.1016/0021-9991(78)90023-2

    Article  MathSciNet  MATH  Google Scholar 

  23. Thomas, D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17, 1030 (1979). https://doi.org/10.2514/3.61273

    Article  MathSciNet  MATH  Google Scholar 

  24. von Neumann, J., Richtmyer, R.D.: A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21, 232–237 (1950). https://doi.org/10.1063/1.1699639

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys. 178, 210–251 (2002). https://doi.org/10.1006/jcph.2002.7041

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, Z.J., Fidkowski, K.J., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H.T., Kroll, M.G., Persson, B., van Leer, V.M.: High-order CFD methods: current status and perspective. Int. J. Num. Methods Fluids 72(8), 811–845 (2013). https://doi.org/10.1002/fld.3767

    Article  MathSciNet  MATH  Google Scholar 

  27. 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. Comput. Phys. 228(21), 8161–8186 (2009). https://doi.org/10.1016/j.jcp.2009.07.036

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, Z.J., Li, Y., Jia, F., Laskowski, G.M., Kopriva, J., Paliath, U., Bhaskaran, R.: Towards industrial large eddy simulation using the FR/CPR method. Comput. Fluids 156(12), 579–589 (2017). https://doi.org/10.1016/j.compfluid.2017.04.026

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, Z., Xu, Y., Wang, W., Hu, R.: Review of shock wave detection method in CFD post-processing. Chin. J. Aeronaut. 26(3), 501–513 (2013). https://doi.org/10.1016/j.cja.2013.05.001

    Article  MATH  Google Scholar 

  30. Zahr, M.J., Persson, P.-O.: An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions. J. Comput. Phys. 365, 105–134 (2018). https://doi.org/10.1016/j.jcp.2018.03.029

    Article  MathSciNet  MATH  Google Scholar 

  31. Zou, D., Bonfiglioli, A., Paciorri, R., Liu, J.: An embedded shock-fitting technique on unstructured dynamic grids. Comput. Fluids 218, 104847 (2021). https://doi.org/10.1016/j.compfluid.2021.104847

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful for the financial support from the AFOSR under grant FA9550-20-1-0315 with Dr. Fariba Fahroo being the Program Manager.

Funding

This study was funded by AFOSR under grant FA9550-20-1-0315.

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Correspondence to Z. J. Wang.

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Wang, Z.J., Fujimoto, T. A Time-Accurate Dynamic Mesh Approach for High-Order Shock Computation. J Sci Comput 102, 5 (2025). https://doi.org/10.1007/s10915-024-02729-5

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