Skip to main content
SpringerLink
Log in

Dynamically adapted grids for interacting discontinuous solutions

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Further development of the dynamic adaptation method for gas dynamics problems that describe multiple interactions of shock waves, rarefaction waves, and contact discontinuities is considered. Using the Woodward-Colella problem and a nonuniformly accelerating piston as examples, the efficiency of the proposed method is demonstrated for the gas dynamics problems with shock wave and contact discontinuity tracking. The grid points are distributed under the control of the diffusion approximation. The choice of the diffusion coefficient for obtaining both quasi-uniform and strongly nonuniform grids for each subdomain of the solution is validated. The interaction between discontinuities is resolved using the Riemann problem for an arbitrary discontinuity. Application of the dynamic adaptation method to the Woodward-Colella problem made it possible to obtain a solution on a grid consisting of 420 cells that is almost identical to the solution obtained using the WENO5m method on a grid consisting of 12 800 cells. In the problem for a nonuniformly accelerating piston, a proper choice of the diffusion coefficient in the transformation functions makes it possible to generate strongly nonuniform grids, which are used to simulate the interaction of a series of shock waves using shock wave and contact discontinuity tracking.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. F. Thompson, Z. U. Z. Warsi, and C. W. Mastin, Numerical Grid Generation (New York, North-Holland, 1985).

    MATH  Google Scholar 

  2. D. A. Anderson, “Equidistribution Schemes, Poisson Generators, and Adaptive Grid,” Appl. Math. Comput. 24, 211–227 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  3. V. D. Liseikin, “Universal Elliptic Method for Adaptive Grid Generation,” Zh. Vychisl. Mat. Mat. Fiz. 44, 2167–2193 (2004) [Comput. Math. Math. Phys. 44, 2062–2086 (2004)].

    MATH  MathSciNet  Google Scholar 

  4. N. A. Dar’in and V. I. Mazhukin, “An Approach to Generation of Adaptive Grids,” Dokl. Akad. Nauk SSSR 298, 64–68 (1988).

    MathSciNet  Google Scholar 

  5. D. A. Anderson, J. C. Tannehill, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer (Hemisphere, New York, 1984; Mir, Moscow, 1990), Vols. 1, 2.

    MATH  Google Scholar 

  6. S. A. Ivanenko and G. P. Prokopov, “Methods of Adaptive Harmonic Grid Generation,” Zh. Vychisl. Mat. Mat. Fiz. 37, 643–662 (1997) [Comput. Math. Math. Phys. 37, 627–645 (1997)].

    MATH  MathSciNet  Google Scholar 

  7. Applied Geometry, Grid Generation, and Highly Accurate Computations, Trudy Vserossiiskoi Konferentsii, Ed. by Yu. G. Evtushenko, M. K. Kerimov, and V. A. Garanzha (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2004) [in Russian].

    Google Scholar 

  8. M. J. Berger and P. Colella, “Local Adaptive Mesh Refinement for Shock Hydrodynamics,” J. Comput. Phys. 82, 64–84 (1989).

    Article  MATH  Google Scholar 

  9. M. J. Berger, “Data Structures for Adaptive Grid Generation,” SIAM J. Sci. Statist. Comput. 3, 904–916 (1986).

    Article  Google Scholar 

  10. J. M. Hyman and Sh. Li, Iterative and Dynamic Control of Udaptive Mesh Refinement with Nested Hierarchical Grids, Los Alamos Lab. Rep, No. 5462 (1998).

  11. A. Andersen, X. Zheng, and V. Cristini, “Adaptive Unstructured Volume Remeshing. I: The Method,” J. Comput. Phys. 208, 616–625 (2005).

    Article  MathSciNet  Google Scholar 

  12. R. R. Nourgaliev, T. N. Dinh, and T. G. Theofanous, “Adaptive Characteristics-Based Matching for Compressible Multifluid Dynamics,” J. Comput. Phys. 213, 500–529 (2006).

    Article  MATH  Google Scholar 

  13. V. I. Mazhukin, A. A. Samarskii, and A. V. Shapranov, “The Dynamic Adaptation Method in the Bürgers Problem,” Dokl. Akad. Nauk 333, 165–169 (1993).

    MathSciNet  Google Scholar 

  14. V. D. Liseikin, “A Survey of Structured Adaptive Grid Generation Methods,” Zh. Vychisl. Mat. Mat. Fiz. 36, 3–41 (1996).

    MathSciNet  Google Scholar 

  15. V. I. Mazhukin, O. Kastel’yanos, A. A. Samarskii, and A. V. Shapranov, “The Dynamic Adaptation Method in Nonstationary Problems with High Gradients,” Mat. Modelir. 5(4), 32–56 (1993).

    MATH  MathSciNet  Google Scholar 

  16. N. A. Dar’in, V. I. Mazhukin, and A. A. Samarskii, “A Finite Difference Method for Solving One-Dimensional Gas Dynamics Problems on Adaptive Grids,” Dokl. Akad. Nauk SSSR 302, 1078–1081 (1988).

    MathSciNet  Google Scholar 

  17. V. D. Liseikin, “Three-Dimensional Grid Generation Technique for Aerogasdynamics: A Survey,” Vopr. Atom. Nauki Tekhn., Ser. Mat. Modelir. Fiz. Protsessov, No. 3, 31–45 (1991).

  18. N. A. Dar’in, V. I. Mazhukin, and A. A. Samarskii, “A Finite Difference Method for Solving One-Dimensional Gas Dynamics Problems on Adaptive Grids that Are Dynamically Related to the Solution,” Zh. Vychisl. Mat. Mat. Fiz. 28, 1210–1225 (1988).

    MATH  MathSciNet  Google Scholar 

  19. P. V. Breslavskii and V. I. Mazhukni, “The Dynamic Adaptation Method in Gas Dynamics,” Mat. Modelir. 7(12), 48–78 (1995).

    Google Scholar 

  20. A. N. Gil’manov, “Application of Dynamically Adaptive Grids to the Analysis of Flows with a Multiscale Structure,” Zh. Vychisl. Mat. Mat. Fiz. 41, 311–326 (2001) [Comput. Math. Math. Phys. 41, 289–303 (2001)].

    MathSciNet  Google Scholar 

  21. D. V. Rudenko and S. V. Utyuzhnikov, “Use of Dynamically Adaptive Grids for Modeling Three-Dimensional Unsteady Gas Flows with High Gradients,” Zh. Vychisl. Mat. Mat. Fiz. 42, 395–409 (2002) [Comput. Math. Math. Phys. 42, 377–390 (2002)].

    MATH  MathSciNet  Google Scholar 

  22. B. N. Azarenok, “A Scheme for Detonation Wave Computation on Moving Meshes,” Zh. Vychisl. Mat. Mat. Fiz. 45, 2260–2282 (2005) [Comput. Math. Math. Phys. 45, 2174–2195 (2005)].

    MATH  MathSciNet  Google Scholar 

  23. A. A. Samarskii and Yu. P. Popov, Finite Difference Methods for Solving Gas Dynamics Problems, 3rd ed. (Nauka, Moscow, 1992) [in Russian].

    Google Scholar 

  24. S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, et al., Numerical Solution of Multidimensional Gas Dynamics Problems (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  25. S. K. Godunov, “A Finite Difference Method for the Numerical Calculation of Discontinuous Solutions to Gas Dynamics Equations,” Mat. Sb. 47(3), 271–306 (1959).

    MathSciNet  Google Scholar 

  26. B. Van Leer, “Towards the Ultimate Conservative Finite Difference Scheme. I. The Quest of Monotonicity,” Lect. Notes Phys. 18(1), 163–168 (1973).

    Article  Google Scholar 

  27. B. Van Leer, “Towards the Ultimate Conservative Finite Difference Scheme. II. Monotonicity and Conservation Combined in a Second Order Scheme,” J. Comput. Phys. 14(4), 361–376 (1974).

    Article  Google Scholar 

  28. J. P. Boris, D. L. Book, and K. Hain, “Flux-Corrected Transport. II. Generalization of the Method,” J. Comput. Phys. 18, 248–283 (1975).

    Article  Google Scholar 

  29. J. P. Boris and D. L. Book, “Flux-Corrected Transport. III Minimal-Error FCT Algorithms,” J. Comput. Phys. 20, 397–431 (1976).

    Article  Google Scholar 

  30. E. S. Oran and J. P. Boris, Numerical Simulation of Reactive Flow (Elsevier, New York, 1987; Mir, Moscow, 1990).

    MATH  Google Scholar 

  31. A. Harten, “High Resolution Schemes for Hyperbolic Conservation Laws,” J. Comput. Phys. 49, 357–393 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  32. A. Harten, “On a Class of High Resolution Total-Variation-Stable Finite-Difference Schemes,” SIAM J. Numer. Anal. 21, 1–23 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  33. S. Osher, “Riemann Solvers, the Entropy Condition, and Difference Approximation,” SIAM J. Numer. Anal. 21, 217–235 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  34. S. Osher and C.-W. Shu, “Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes,” J. Comput. Phys. 77, 439–471 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Harten, “ENO Schemes with Subcell Resolution,” J. Comput. Phys. 83, 148–184 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  36. X.-D. Liu, S. Osher, and T. Chan, “Weighted Essentially Non-Oscillatory Schemes,” J. Comput. Phys. 115, 200–212 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  37. Ph. Colella and P. R. Woodward, “The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations,” J. Comput. Phys. 54, 174–201 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  38. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Issues in the Numerical Solution of Hyperbolic Equations (Fizmatlit, Moscow, 2001) [in Russian].

    Google Scholar 

  39. Yu. A. Bondarenko, V. V. Bashurov, and Yu. V. Yanilkin, Mathematical Models and Numerical Methods for Unsteady Gas Dynamics Problems: A Survey, Preprint, No. 88, RFYaTs-VNIIEF (2003).

  40. R. J. LeVeque and K. M. Shyue, “One-Dimensional Front Tracking Based on High Resolution Wave Propagation Methods,” SISC 16, 348–377 (1995).

    MATH  MathSciNet  Google Scholar 

  41. B. Van Leer, “Towards the Ultimate Conservative Difference Scheme. V. A Second Order Sequel to Godunov’s Method,” J. Comput. Phys. 32, 101–136 (1979).

    Article  Google Scholar 

  42. W. D. Henshaw, “A Scheme for Numerical Solution of Hyperbolic Systems of Conservation Laws,” J. Comput. Phys. 68, 25–47 (1987).

    Article  MATH  Google Scholar 

  43. K. M. Shuye, “An Efficient Shock-Capturing Algorithm for Compressible Multicomponent Problems,” J. Comput. Phys. 142, 208–242 (1998).

    Article  MathSciNet  Google Scholar 

  44. S. A. Ivanenko and A. A. Charakhch’yan, “Curved Grids Consisting of Convex Quadrangles,” Zh. Vychisl. Mat. Mat. Fiz. 28, 503–514 (1988).

    MATH  MathSciNet  Google Scholar 

  45. V. D. Liseikin, “On Some Interpretations of a Smoothness Functional Used in Constructing Regular and Adaptive Grids,” Russ. J. Numer. Anal. Modell. 8, C. 507–518 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  46. P. V. Breslavskii and V. I. Mazhukin, “Mathematical Modeling of Pulse Melting and Evaporation of Metals with the Explicit Tracking of the Interphase Boundaries,” Inzh.-Fiz. Zhurnal 57(1), 107–114 (1989).

    Google Scholar 

  47. L. V. Ovsyannikov, Lectures in the Foundations of Gas Dynamics (Izhevsk, Institut Komputernykh Issledovanii, 2003) [in Russian].

    Google Scholar 

  48. O. N. Koroleva and V. I. Mazhukin, “Mathematical Simulation of Laser Induced Melting and Evaporation of Multilayer Materials,” Zh. Vychisl. Mat. Mat. Fiz. 46, 887–901 (2006) [Comput. Math. Math. Phys. 46, 848–862 (2006)].

    MathSciNet  Google Scholar 

  49. A. K. Henrick, T. D. Aslam, and J. M. Powers, “Mapped Weighted Essentially Non-Oscillatory Schemes: Achieving Optimal Order Near Critical Points,” J. Comput. Phys. 207, 542–567 (2005).

    Article  MATH  Google Scholar 

  50. V. I. Mazhukin, I. Smurov, C. Dupuy, and D. Jeandel, “Simulationn of Laser Induced Melting and Evaporation Processes in Superconducting Ceramics,” J. Numer. Heat Transfer 26Part A, 587–600 (1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematical Modeling, Miusskaya pl. 4, Moscow, 125047, Russia

    P. V. Breslavskii & V. I. Mazhukin

Authors
  1. P. V. Breslavskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. I. Mazhukin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © P.V. Breslavskii, V I. Mazhukin, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 4, pp. 717–737.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Breslavskii, P.V., Mazhukin, V.I. Dynamically adapted grids for interacting discontinuous solutions. Comput. Math. and Math. Phys. 47, 687–706 (2007). https://doi.org/10.1134/S0965542507040124

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542507040124

Keywords

Search

Navigation