A Cartesian grid method for the three-dimensional numerical simulation of shock wave propagation in complex-shape domains with moving boundaries


  • V.V. Elesin Institute for Design Automation of RAS (IAP RAS)
  • D.A. Sidorenko Institute for Design Automation of RAS (IAP RAS)
  • P.S. Utkin Institute for Design Automation of RAS (IAP RAS)




mathematical modeling, three-dimensional Euler equations, Cartesian grid method, shock wave


This paper is devoted to the development and quantitative estimation of a numerical algorithm based on the Cartesian grid method for the three-dimensional mathematical simulation of shock wave propagation in domains of complex varying shapes. A detailed description of the numerical algorithm is presented. Its key element is the specification of numerical fluxes through the edges that are common for the inner regular cells of the computational domain and the outer cells intersected by the boundaries of the bodies. The efficiency of the algorithm is shown by comparing the numerical and experimental data in the problems of interaction of a shock wave with a fixed sphere and a moving particle.

Author Biographies

V.V. Elesin

D.A. Sidorenko

P.S. Utkin


  1. S. V. Dyachenko, “Development of a Software Package for 3D Modeling of Multiphase Multicomponent Flows in Nuclear Power Engineering,” Vychisl. Metody Programm. 15, 162-182 (2014).
  2. A. V. Glazunov, “Numerical Simulation of Turbulence and Transport of Fine Particulate Impurities in Street Canyons,” Vychisl. Metody Programm. 19, 17-37 (2018).
  3. K. N. Volkov, V. N. Emelyanov, and I. V. Teterina, “Visualization of Numerical Results Obtained for Gas-Particle Flows Using Lagrangian Approaches to the Dispersed Phase Description,” Vychisl. Metody Programm. 19, 522-539 (2018).
  4. A. A. Fedorov, “Droplet Visualization in FlowVision,” Vychisl. Metody Programm. 19, 1-8 (2018).
  5. I. A. Bedarev and A. V. Fedorov, “Direct Simulation of the Relaxation of Several Particles Behind Transmitted Shock Waves,” Inzh. Fiz. Zh. 90 (2), 450-457 (2017) [J. Eng. Phys. Thermophys. 90 (2), 423-429 (2017)].
  6. O. Sen, N. J. Gaul, K. K. Choi, et al., “Evaluation of Kriging Based Surrogate Models Constructed from Mesoscale Computations of Shock Interaction with Particles,” J. Comput. Phys. 336, 235-260 (2017).
  7. S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, et al., Numerical Solution of Multidimensional Gas Dynamics Problems (Nauka, Moscow, 1976) [in Russian].
  8. R. Mittal and G. Iaccarino, “Immersed Boundary Methods,” Annu. Rev. Fluid Mech. 37, 239-261 (2005).
  9. W. P. Bennett, N. Nikiforakis, and R. Klein, “A Moving Boundary Flux Stabilization Method for Cartesian Cut-Cell Grids Using Directional Operator Splitting,” J. Comput. Phys. 368, 333-358 (2018).
  10. R. B. Pember, J. B. Bell, P. Colella, et al., “An Adaptive Cartesian Grid Method for Unsteady Compressible Flow in Irregular Regions,” J. Comput. Phys. 120 (2), 278-304 (1995).
  11. P. Colella, D. T. Graves, B. J. Keen, and D. Modiano, “A Cartesian Grid Embedded Boundary Method for Hyperbolic Conservation Laws,” J. Comput. Phys. 211 (1), 347-366 (2006).
  12. X. Y. Hu, B. C. Khoo, N. A. Adams, and F. L. Huang, “A Conservative Interface Method for Compressible Flows,” J. Comput. Phys. 219 (2), 553-578 (2006).
  13. L. Schneiders, D. Hartmann, M. Meinke, and W. Schröder, “An Accurate Moving Boundary Formulation in Cut-Cell Methods,” J. Comput. Phys. 235, 786-809 (2013).
  14. P. Colella, “Multidimensional Upwind Methods for Hyperbolic Conservation Laws,” J. Comput. Phys. 87 (1), 171-200 (1990).
  15. R. Klein, K. R. Bates, and N. Nikiforakis, “Well-Balanced Compressible Cut-Cell Simulation of Atmospheric Flow,” Phil. Trans. Roy. Soc. Ser. A. Math. Phys. Eng. Sci. 367, 4559-4575 (2009).
  16. L. Schneiders, C. Günther, M. Meinke, and W. Schröder, “An Efficient Conservative Cut-Cell Method for Rigid Bodies Interacting with Viscous Compressible Flows,” J. Comput. Phys. 311, 62-86 (2016).
  17. D. K. Clarke, H. A. Hassan, and M. D. Salas, “Euler Calculations for Multielement Airfoils Using Cartesian Grids,” AIAA J. 24 (3), 353-358 (1986).
  18. J. J. Quirk, “An Alternative to Unstructured Grids for Computing Gas Dynamic Flows around Arbitrarily Complex Two-Dimensional Bodies,” Comput. Fluids 23 (1), 125-142 (1994).
  19. M. J. Berger, C. Helzel, and R. J. LeVeque, “H-box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids,” SIAM J. Numer. Anal. 41 (3), 893-918 (2003).
  20. D. M. Ingram, D. M. Causon, and C. G. Mingham, “Developments in Cartesian Cut Cell Methods,” Math. Comput. Simul. 61 (3-6), 561-572 (2003).
  21. S. Xu, T. Aslam, and D. S. Stewart, “High Resolution Numerical Simulation of Ideal and Non-ideal Compressible Reacting Flows with Embedded Internal Boundaries,” Combust. Theory Model. 1 (1), 113-142 (1997).
  22. G. Yang, D. M. Causon, D. M. Ingram, et al., “A Cartesian Cut Cell Method for Compressible Flows. Part A: Static Body Problems,” Aeronaut. J. 101 (1002), 47-56 (1997).
  23. P. T. Barton, B. Obadia, and D. Drikakis, “A Conservative Level-Set Based Method for Compressible Solid/Fluid Problems on Fixed Grids,” J. Comput. Phys. 230 (21), 7867-7890 (2011).
  24. D. Hartmann, M. Meinke, and W. Schröder, “A Strictly Conservative Cartesian Cut-Cell Method for Compressible Viscous Flows on Adaptive Grids,” Comput. Methods Appl. Mech. Eng. 200 (9-12), 1038-1052 (2011).
  25. A. Pogorelov, M. Meinke, and W. Schröder, “Cut-Cell Method Based Large-Eddy Simulation of Tip-Leakage Flow,” Phys. Fluids 27 (2015).
    doi 10.1063/1.4926515
  26. A. Pogorelov, M. Meinke, and W. Schröder, “Effects of Tip-Gap Width on the Flow Field in an Axial Fan,” Int. J. Heat Fluid Fl. 61, 466-481 (2016).
  27. A. Pogorelov, L. Schneiders, M. Meinke, and W. Schröder, “An Adaptive Cartesian Mesh Based Method to Simulate Turbulent Flows of Multiple Rotating Surfaces,” Flow Turbul. Combust. 100 (1), 19-38 (2018).
  28. C. Helzel, M. J. Berger, and R. J. LeVeque, “A High-Resolution Rotated Grid Method for Conservation Laws with Embedded Geometries,” SIAM J. Sci. Comput. 26 (3), 785-809 (2005).
  29. M. Berger and C. Helzel, “A Simplified h-box Method for Embedded Boundary Grids,” SIAM J. Sci. Comput. 34 (2), A861-A888 (2012).
  30. D. A. Sidorenko and P. S. Utkin, “A Cartesian Grid Method for the Numerical Modeling of Shock Wave Propagation in Domains of Complex Shape,” Vychisl. Metody Programm. 17, 353-364 (2016).
  31. D. A. Sidorenko and P. S. Utkin, “Two-Dimensional Gas-Dynamic Modeling of the Interaction of a Shock Wave with Beds of Granular Media,” Khim. Fiz. 37 (9), 43-49 (2018) [Russ. J. Phys. Chem. B. 12 (5), 869-874 (2018)].
  32. D. A. Sidorenko and P. S. Utkin, “Numerical Modeling of the Relaxation of a Body behind the Transmitted Shock Wave,” Mat. Model. 30 (11), 91-104 (2018). [Math. Models Comput. Simul. 11 (4), 509-517 (2019)].
  33. A. Chertock and A. Kurganov, “A Simple Eulerian Finite-Volume Method for Compressible Fluids in Domains with Moving Boundaries,” Commun. Math. Sci. 6 (3), 531-556 (2008).
  34. J. L. Steger and R. F. Warming, “Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite-Difference Methods,” J. Comput. Phys. 40 (2), 263-293 (1981).
  35. M. Pandolfi and D. D’Ambrosio, “Numerical Instabilities in Upwind Methods: Analysis and Cures for the ’Carbuncle’ Phenomenon,” J. Comput. Phys. 166 (2), 271-301 (2001).
  36. H. Tanno, K. Itoh, T. Saito, et al., “Interaction of a Shock with a Sphere Suspended in a Vertical Shock Tube,” Shock Waves 13 (3), 191-200 (2003).
  37. V. M. Boiko, A. V. Fedorov, V. M. Fomin, et al., “Ignition of Small Particles Behind Shock Waves,” in Shock Waves, Explosions and Detonations (American Inst. of Aeronautics and Astronautics, New York, 1983), pp. 71-87.



How to Cite

Елесин В., Сидоренко Д., Уткин П. A Cartesian Grid Method for the Three-Dimensional Numerical Simulation of Shock Wave Propagation in Complex-Shape Domains With Moving Boundaries // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2019. 20. 309-322. doi 10.26089/NumMet.v20r327



Section 1. Numerical methods and applications