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.
Similar content being viewed by others
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
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
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
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
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
Dahl, M.D. Ed.: Third computational aeroacoustics (CAA) workshop on benchmark problems, technical report NASA/CP-2000-209790, NASA (2000).
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
Gnoffo, P., White, J.A.: Computational aerothermodynamic simulation issues on unstructured grids. AIAA-2004–2371; 2004 https://doi.org/10.2514/6.2004-2371
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
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
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
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
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
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
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
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
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
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
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)
Richtmyer, R.D.: A proposed method for the calculation of shocks. Technical Report LA–671, Los Alamos Scientific Laboratory (1948)
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
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
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
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
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
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
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
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
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
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
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
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
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.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-024-02729-5