Comparison of a modified large-particle method with some high resolution schemes. Two-Dimensional test problems

Authors

DOI:

https://doi.org/10.26089/NumMet.v20r329

Keywords:

large-particle method, high resolution, test problems, computational properties

Abstract

A number of computational properties of the previously proposed new modification of a large-particle method are studied on the basis of a nonlinear correction of artificial viscosity at the first (Eulerian) stage and a hybridization of fluxes at the second (Lagrangian and final) stage supplemented by a two-step Runge-Kutta algorithm in time. The method has a second order of approximation in space and time on smooth solutions. The computational efficiency of the method is shown compared to several modern high resolution schemes using the forward facing step problem and the double Mach reflection problem.

Author Biographies

D.V. Sadin

B.V. Belyaev

A.F. Mozhaysky Military Space Academy
• Associate Professor

V.A. Davidchuk

References

  1. P. Colella and P. R. Woodward, “The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations,” J. Comput. Phys. 54 (1), 174-201 (1984).
  2. G.-S. Jiang and C.-W. Shu, “Efficient Implementation of Weighted ENO Schemes,” J. Comput. Phys. 126 (1), 202-228 (1996).
  3. B. Cockburn and C.-W. Shu, “Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems,” J. Sci. Comput. 16 (3), 173-261 (2001).
  4. F. Kemm, “On the Proper Setup of the Double Mach Reflection as a Test Case for the Resolution of Gas Dynamics Codes,” Comput. Fluids 132, 72-75 (2016).
  5. I. Yu. Tagirova and A. V. Rodionov, “Application of Artificial Viscosity for Suppressing the Carbuncle Phenomenon in Godunov-Type Schemes,” Mat. Model. 27 (10), 47-64 (2015) [Math. Models Comput. Simul. 8 (3), 249-262 (2016)].
  6. D. V. Sadin, “TVD Scheme for Stiff Problems of Wave Dynamics of Heterogeneous Media of Nonhyperbolic Nonconservative Type,” Zh. Vychisl. Mat. Mat. Fiz. 56 (12), 2098-2109 (2016) [Comput. Math. Math. Phys. 56 (12), 2068-2078 (2016)].
  7. D. V. Sadin, “Schemes with Customizable Dissipative Properties as Applied to Gas-Suspensions Flow Simulation,” Mat. Model. 29 (12), 89-104 (2017).
  8. D. V. Sadin, “Application of Scheme with Customizable Dissipative Properties for Gas Flow Calculation with Interface Instability Evolution,” Nauch.-Tekhn. Vestn. Inform. Tekhnol. Mekhan. Optik. 18 (1), 153-157 (2018).
  9. D. V. Sadin and V. A. Davidchuk, “Comparison of a Modified Large-Particle Method with Some High Resolution Schemes. One-Dimensional Test Problems,” Vychisl. Metody Programm. 20, 138-146 (2019).
  10. D. V. Sadin, S. D. Lyubarskii, and Yu. A. Gravchenko, “Features of an Underexpanded Pulsed Impact Gas-Dispersed Jet with a High Particle Concentration,” Zh. Tekh. Fiz. 87 (1), 22-26 (2017) [Tech. Fiz. 62 (1), 18-23 (2017)].
  11. V. M. Goloviznin, S. A. Karabasov, and V. G. Kondakov, “Generalization of the CABARET Scheme to Two-Dimensional Orthogonal Computational Grids,” Mat. Model. 25 (7), 103-136 (2013) [Math. Models Comput. Simul. 6 (1), 56-79 (2014)].
  12. R. Landshoff,  A Numerical Method for Treating Fluid Flow in the Presence of Shocks , Technical Report LA-1930 (Los Alamos Nat. Lab., Los Alamos, 1955).
  13. R. B. Christensen, Godunov Methods on a Staggered Mesh - An Improved Artificial Viscosity , Preprint UCRL-JC-105269 (Lawrence Livermore Nat. Lab., Livermore, 1990).
  14. A. F. Emery, “An Evaluation of Several Differencing Methods for Inviscid Fluid Flow Problems,” J. Comput. Phys. 2 (3), 306-331 (1968).
  15. X. Liu, S. Zhang, H. Zhang, and C.-W. Shu, “A New Class of Central Compact Schemes with Spectral-Like Resolution II: Hybrid Weighted Nonlinear Schemes,” J. Comput. Phys. 284, 133-154 (2015).
  16. P. V. Bulat and K. N. Volkov, “Simulation of Supersonic Flow in a Channel with a Step on Nonstructured Meshes with the Use of the Weno Scheme,” Inzh. Fiz. Zh. 88 (4), 848-855 (2015) [J. Eng. Phys. Thermophys. 88 (6), 1582-1582 (2015)]
  17. S. A. Isaev and D. A. Lysenko, “Testing of Numerical Methods, Convective Schemes, Algorithms for Approximation of Flows, and Grid Structures by the Example of a Supersonic Flow in a Step-Shaped Channel with the Use of the CFX and Fluent Packages,” Inzh. Fiz. Zh. 82 (2), 326-330 (2009) [J. Eng. Phys. Thermophys. 82 (2), 321-326 (2009)].
  18. M. P. Galanin and E. B. Savenkov, Numerical Analysis Methods for Mathematical Models (Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Moscow, 2010) [in Russian].
  19. I. V. Popov and I. V. Fryazinov, “Calculations of Two-Dimensional Test Problems by the Method of Adaptive Viscosity,” Mat. Model. 22 (5), 57-66 (2010) [Math. Models Comput. Simul. 2 (6), 724-732 (2010)].
  20. M. P. Galanin, E. B. Savenkov, and S. A. Tokareva, “Solving Gas Dynamics Problems with Shock Waves using the Runge-Kutta Discontinuous Galerkin Method,” Mat. Model. 20 (11), 55-66 (2008). [Math. Models Comput. Simul. 1 (5), 635-645 (2009)].
  21. A. N. Semenov, M. K. Berezkina, and I. V. Krasovskaya, “Classification of Shock Wave Reflections from a Wedge. Part 2: Experimental and Numerical Simulations of Different Types of Mach Reflections,” Zh. Tekh. Fiz. 79 (4), 52-58 (2009) [Tech. Phys. 54 (4), 497-503 (2009)].
  22. J. Shi, Y.-T. Zhang, and C.-W. Shu, “Resolution of High Order WENO Schemes for Complicated Flow Structures,” J. Comput. Phys. 186 (2), 690-696 (2003).
  23. N. M. Evstigneev, “On the Construction and Properties of WENO-Schemes Order Five, Seven, Nine, Eleven and Thirteen. Part 2. Numerical Examples,” Comput. Res. Model. 8 (6), 885-910 (2016).
  24. R. Liska and B. Wendroff, “Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations,” SIAM J. Sci. Comput. 25 (3), 995-1017 (2003).

Published

2019-10-29

How to Cite

Садин Д.В., Беляев Б.В., Давидчук В.А. Comparison of a Modified Large-Particle Method With Some High Resolution Schemes. Two-Dimensional Test Problems // Numerical methods and programming. 2019. 20. 337-345. doi 10.26089/NumMet.v20r329

Issue

Section

Section 1. Numerical methods and applications