Конструирование схем третьего порядка точности с помощью разложений Лагранжа-Бюрмана для численного интегрирование уравнений невязкого газа
Ключевые слова:
гиперболические законы сохранения
разложения Лагранжа-Бюрмана
разностные методы
Аннотация
Предлагается строить явные разностные схемы третьего порядка точности для гиперболических законов сохранения с применением разложений сеточных функций в ряды Лагранжа-Бюрмана. Результаты тестовых расчетов для случаев одномерного уравнения переноса и многомерных уравнений Эйлера невязкого сжимаемого газа подтверждают третий порядок точности построенных схем. Получены квазимонотонные профили численных решений.
Раздел
Раздел 1. Вычислительные методы и приложения
Библиографические ссылки
- A. A. Samarskii and Yu. P. Popov, Difference Schemes of Gas Dynamics (Nauka, Moscow, 1980) [in Russian].
- A. I. Tolstykh, Compact Difference Schemes and Their Application to Aerohydrodynamic Problems (Nauka, Moscow, 1990) [in Russian].
- R. J. LeVeque, Numerical Methods for Conservation Laws (Birkh854user, Basel, 1992).
- A. I. Tolstykh, High Accuracy Non-Centered Compact Difference Schemes for Fluid Dynamics Applications (World Scientific, Singapore, 1994).
- V. I. Pinchukov and C.-W. Shu, High Order Numerical Methods for the Problems of Aerodynamics (Izd. Ross. Akad. Nauk, Novosibirsk, 2000) [in Russian].
- E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (Springer, Berlin, 2010).
- 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].
- 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).
- R. Richtmyer and K. Morton, Difference Methods for Initial Value Problems (Wiley, New York, 1967; Mir, Moscow, 1972).
- 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).
- I. V. Popov and I. V. Fryazinov, Method of Adaptive Artificial Viscosity for Solving the Gas Dynamics Equations (Krasand, Moscow, 2014) [in Russian].
- R. J. LeVeque, Finite-Volume Methods for Hyperbolic Problems (Cambridge Univ. Press, Cambridge, 2004).
- E. V. Vorozhtsov, Exercises for the Theory of Difference Schemes (Novosibirsk. Tekh. Univ., Novosibirsk, 2000) [in Russian].
- 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
- 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)].
- P. D. Lax and B. Wendroff, “Systems of Conservation Laws III,” Comm. Pure Appl. Math. 13 (2), 217-237 (1960).
- R. W. MacCormack, “The Effect of Viscosity in Hypervelocity Impact Cratering,” AIAA Paper (1969).
doi 10.2514/6.1969-354
- 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)].
- S. Z. Burstein and A. A. Mirin, “Third Order Difference Methods for Hyperbolic Equations,” J. Comput. Phys. 5 (3), 547-571 (1970).
- 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)].
- 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).
- 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].
- E. V. Vorozhtsov and N. N. Yanenko, Methods for the Localization of Singularities in Numerical Solutions of Gas Dynamics Problems (Springer, New York, 1990).
- 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).
- 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
- S. Osher and S. R. Chakravarthy, Very High Order Accurate TVD Schemes , ICASE Report No. 84-44 (1984).
- 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.
- S. R. Chakravarthy and K.-Y. Szema, “Euler Solver for Three-Dimensional Supersonic Flows with Subsonic Pockets,” J. Aircraft 24 (2), 73-83 (1987).
- 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).
- 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).
- 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).
- 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).
- 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.
- E. V. Vorozhtsov, “Application of Lagrange-Bürmann Expansions for the Numerical Integration of the Inviscid Gas Equations,” Vychisl. Metody Programm. 12, 348-361 (2011).
- 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).
- 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.
- W. Strampp, V. Ganzha, and E. Vorozhtsov, Höhere Mathematik mit Mathematica (Vieweg, Braunschweig, 1997).
- 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).
- 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).
- V. G. Ganzha and E. V. Vorozhtsov, Computer-Aided Analysis of Difference Schemes for Partial Differential Equations (Wiley, New York, 1996).
- 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).
- 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).
- M. S. Liou and C. J. Steffen, “A New Flux Splitting Scheme,” J. Comput. Phys. 107 (1), 23-39 (1993).
- 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).
- P. K. Sweby, “High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws,” SIAM J. Num. Anal. 21 (5), 995-1011 (1984).
- S. F. Davis, “A Simplified TVD Finite Difference Scheme via Artificial Viscosity,” SIAM J. Sci. and Statist. Comput. 8 (1), 1-18 (1987).
- S. Osher and S. Chakravarthy, “High Resolution Schemes and the Entropy Condition,” SIAM J. Num. Anal. 21 (5), 984-995 (1984).
- 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).
- 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).