Construction of third-order schemes using Lagrange-Bürmann expansions for the numerical integration of inviscid gas equations

Authors

  • E.V. Vorozhtsov S.A. Khristianovich Institute of Theoretical and Applied Mechanics of SB RAS

DOI:

https://doi.org/10.26089/NumMet.v17r104

Keywords:

hyperbolic conservation laws, Lagrange-Burmann expansions, difference methods

Abstract

It is proposed to construct several explicit third-order difference schemes for the hyperbolic conservation laws using the expansions of grid functions in Lagrange-Burmann series. The results of test computations for the one-dimensional advection equation and multidimensional Euler equations governing the inviscid compressible gas flows confirm the third order of accuracy of the constructed schemes. The quasi-monotonous profiles of numerical solutions are obtained.

Author Biography

E.V. Vorozhtsov

S.A. Khristianovich Institute of Theoretical and Applied Mechanics of SB RAS
• Leading Researcher

References

  1. A. A. Samarskii and Yu. P. Popov, Difference Schemes of Gas Dynamics (Nauka, Moscow, 1980) [in Russian].
  2. A. I. Tolstykh, Compact Difference Schemes and Their Application to Aerohydrodynamic Problems (Nauka, Moscow, 1990) [in Russian].
  3. R. J. LeVeque, Numerical Methods for Conservation Laws (Birkh854user, Basel, 1992).
  4. A. I. Tolstykh, High Accuracy Non-Centered Compact Difference Schemes for Fluid Dynamics Applications (World Scientific, Singapore, 1994).
  5. V. I. Pinchukov and C.-W. Shu, High Order Numerical Methods for the Problems of Aerodynamics (Izd. Ross. Akad. Nauk, Novosibirsk, 2000) [in Russian].
  6. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (Springer, Berlin, 2010).
  7. K. N. Volkov, Yu. N. Deryugin, V. N. Emel’yanov, A. S. Kozelkov, and I. V. Teterina, Difference Schemes in Gas Dynamics on Unstructured Grids (Fizmatlit, Moscow, 2014) [in Russian].
  8. W. Boscheri, D. S. Balsara, and M. Dumbser, “Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based on Genuinely Multidimensional HLL Riemann Solvers,” J. Comput. Phys. 267, 112-138 (2014).
  9. R. Richtmyer and K. Morton, Difference Methods for Initial Value Problems (Wiley, New York, 1967; Mir, Moscow, 1972).
  10. B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and their Application to Gas Dynamics (Nauka, Moscow, 1978; Amer. Math. Soc., Providence, 1983).
  11. I. V. Popov and I. V. Fryazinov, Method of Adaptive Artificial Viscosity for Solving the Gas Dynamics Equations (Krasand, Moscow, 2014) [in Russian].
  12. R. J. LeVeque, Finite-Volume Methods for Hyperbolic Problems (Cambridge Univ. Press, Cambridge, 2004).
  13. E. V. Vorozhtsov, Exercises for the Theory of Difference Schemes (Novosibirsk. Tekh. Univ., Novosibirsk, 2000) [in Russian].
  14. A. Jameson, W. Schmidt, and E. Turkel, “Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,” AIAA Paper (1981).
    doi 10.2514/6.1981-1259
  15. M. V. Lipavskii and A. I. Tolstykh, “Tenth-Order Accurate Multioperator Scheme and Its Application in Direct Numerical Simulation,” Zh. Vychisl. Mat. Mat. Fiz. 53 (4), 600-614 (2013) [Comput. Math. Math. Phys. 53 (4), 455-468 (2013)].
  16. P. D. Lax and B. Wendroff, “Systems of Conservation Laws III,” Comm. Pure Appl. Math. 13 (2), 217-237 (1960).
  17. R. W. MacCormack, “The Effect of Viscosity in Hypervelocity Impact Cratering,” AIAA Paper (1969).
    doi 10.2514/6.1969-354
  18. V. V. Rusanov, “Difference Schemes of the Third-Order Accuracy for Continuous Computation of Discontinuous Solutions,” Dokl. Akad. Nauk SSSR 180 (6), 1303-1305 (1968) [Sov. Math. Dokl. 9, 771-777 (1968)].
  19. S. Z. Burstein and A. A. Mirin, “Third Order Difference Methods for Hyperbolic Equations,” J. Comput. Phys. 5 (3), 547-571 (1970).
  20. V. B. Balakin, “Methods of the Runge-Kutta Type for Gas Dynamics,” Zh. Vychisl. Mat. Mat. Fiz. 10 (6), 1512-1519 (1970) [USSR Comput. Math. Math. Phys. 10 (6), 208-216 (1970)].
  21. R. F. Warming, P. Kutler, and H. Lomax, “Second- and Third-Order Noncentered Difference Schemes for Nonlinear Hyperbolic Equations,” AIAA J. 11 (2), 189-196 (1973).
  22. E. V. Vorozhtsov and N. N. Yanenko, Methods for the Localization of Singularities in Numerical Solutions of Gas Dynamics Problems (Nauka, Novosibirsk, 1985) [in Russian].
  23. E. V. Vorozhtsov and N. N. Yanenko, Methods for the Localization of Singularities in Numerical Solutions of Gas Dynamics Problems (Springer, New York, 1990).
  24. N. J. Zabusky, S. Gupta, and Y. Gulak, “Localization and Spreading of Contact Discontinuity Layers in Simulations of Compressible Dissipative Flows,” J. Comput. Phys. 188 (2), 348-364 (2003).
  25. S. R. Chakravarthy and S. Osher, “A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws,” AIAA Paper (1985).
    doi 10.2514/6.1985-0363
  26. S. Osher and S. R. Chakravarthy, Very High Order Accurate TVD Schemes , ICASE Report No. 84-44 (1984).
  27. S. Osher and S. R. Chakravarthy, “Very High Order Accurate TVD Schemes,” in IMA Volumes in Mathematics and its Applications (Springer, Heidelberg, 1986), Vol. 2, pp. 229-274.
  28. S. R. Chakravarthy and K.-Y. Szema, “Euler Solver for Three-Dimensional Supersonic Flows with Subsonic Pockets,” J. Aircraft 24 (2), 73-83 (1987).
  29. S. Yamamoto and H. Daiguji, “Higher-Order-Accurate Upwind Schemes for Solving the Compressible Euler and Navier-Stokes Equations,” Comp. Fluids 22 (2-3), 259-270 (1993).
  30. H. Daiguji, X. Yuan, and S. Yamamoto, “Stabilization of Higher-Order High Resolution Schemes for the Compressible Navier-Stokes Equations,” Int. J. Numer. Methods Heat & Fluid Flow 7 (2/3), 250-274 (1997).
  31. C. Bona, C. Bona-Casas, and J. Terradas, “Linear High-Resolution Schemes for Hyperbolic Conservation Laws: TVB Numerical Evidence,” J. Comput. Phys. 228 (6), 2266-2281 (2009).
  32. A. Liska and B. Wendroff, “Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations,” SIAM J. Sci. Comput. 25, 995-1017 (2003).
  33. E. V. Vorozhtsov, “Construction of Difference Schemes for Hyperbolic Conservation Laws with the Aid of the Lagrange-Bürmann Expansions,” in Proc. Int. Conf. on Numerical Mathematics, Novosibirsk, Russia, June 21-25, 2004 (Price-Courier Pres, Novosibirsk, 2004), Part 1, pp. 443-448.
  34. E. V. Vorozhtsov, “Application of Lagrange-Bürmann Expansions for the Numerical Integration of the Inviscid Gas Equations,” Vychisl. Metody Programm. 12, 348-361 (2011).
  35. E. V. Vorozhtsov, “Derivation of Explicit Difference Schemes for Ordinary Differential Equations with the Aid of Lagrange-Bürmann Expansions,” Vychisl. Metody Programm. 11, 198-209 (2010).
  36. E. V. Vorozhtsov, “Derivation of Explicit Difference Schemes for Ordinary Differential Equations with the Aid of Lagrange-Bürmann Expansions,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2010), Vol. 6244, pp. 250-266.
  37. W. Strampp, V. Ganzha, and E. Vorozhtsov, Höhere Mathematik mit Mathematica (Vieweg, Braunschweig, 1997).
  38. B. van  Leer, “Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov’s Method,” J. Comput. Phys. 32 (1), 101-136 (1979).
  39. J. Pike and P. L. Roe, “Accelerated Convergence of Jameson’s Finite-Volume Euler Scheme Using van der Houwen Integrators,” Comp. Fluids 13 (2), 223-236 (1985).
  40. V. G. Ganzha and E. V. Vorozhtsov, Computer-Aided Analysis of Difference Schemes for Partial Differential Equations (Wiley, New York, 1996).
  41. J. L. Steger and R. F. Warming, “Flux Vector Splitting of the Inviscid Gasdynamic Equations with Applications to Finite Difference Methods,” J. Comput. Phys. 40, 263-293 (1981).
  42. W. K. Anderson, J. L. Thomas, and B. van Leer, “Comparison of Finite Volume Flux Vector Splittings for the Euler Equations,” AIAA J. 24 (9), 1453-1460 (1986).
  43. M. S. Liou and C. J. Steffen, “A New Flux Splitting Scheme,” J. Comput. Phys. 107 (1), 23-39 (1993).
  44. E. F. Toro, C. E. Castro, and B. J. Lee, “A Novel Numerical Flux for the 3D Euler Equations with General Equation of State,” J. Comput. Phys. 303, 80-94 (2015).
  45. P. K. Sweby, “High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws,” SIAM J. Num. Anal. 21 (5), 995-1011 (1984).
  46. S. F. Davis, “A Simplified TVD Finite Difference Scheme via Artificial Viscosity,” SIAM J. Sci. and Statist. Comput. 8 (1), 1-18 (1987).
  47. S. Osher and S. Chakravarthy, “High Resolution Schemes and the Entropy Condition,” SIAM J. Num. Anal. 21 (5), 984-995 (1984).
  48. K. Wu, Z. Yang, and H. Tang, “A Third-Order Accurate Direct Eulerian GRP Scheme for the Euler Equations in Gas Dynamics,” J. Comput. Phys. 264, 177-208 (2014).
  49. M. Lahooti and A. Pishevar, “A New Fourth Order Central WENO Method for 3D Hyperbolic Conservation Laws,” Appl. Math. Comput. 218 (20), 10258-10270 (2012).
New computational technologies

Downloads

Published

06-02-2016

How to Cite

Ворожцов Е. Construction of Third-Order Schemes Using Lagrange-Bürmann Expansions for the Numerical Integration of Inviscid Gas Equations // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2016. 17. 21-43. doi 10.26089/NumMet.v17r104

Issue

Vol. 17 (2016): Issue 1.

Section

Section 1. Numerical methods and applications

Most read articles by the same author(s)