Numerical modeling of a two-point correlator for the Lagrange solutions of some evolution equations




equations with random coefficients, intermittency, statistical moment


This paper is devoted to the two-point moments of the solutions arising in simple Lagrange models for the induction equations in the case of finite correlation time of a random medium. We consider the question on the connection between the commutative properties of the corresponding algebraic operators and the minimal sample size of independent random realizations necessary in numerical experiments for modeling the two-point correlator of the solution. It is shown that, as for the one-point moments, the numerical study of the two-point correlator in the case of commutating operators (random numbers) requires a much smaller sample size than in the case when they do not commute (random matrices).

Author Biographies

D.A. Grachev

E.A. Mikhailov


  1. Ya. B. Zel’dovich, S. A. Molchanov, A. A. Ruzmaikin, and D. D. Sokolov, “Intermittency in Random Media,” Usp. Fiz. Nauk 152 (1), 3-32 (1987) [Sov. Phys. Usp. 30 (5), 353-369 (1987)].
  2. Ya. B. Zel’dovich, A. A. Ruzmaikin, and D. D. Sokoloff, The Almighty Chance (World Scientific, Singapore, 1990).
  3. Ya. B. Zel’dovich, “Observations in a Universe Homogeneous in the Mean,” Astron. Zh. 41 (1), 19-24 (1964) [Sov. Astron. 8 (1), 13-16 (1964)].
  4. M. E. Artyushkova and D. D. Sokoloff, “Numerical Modeling of Conjugated Point Distribution along a Geodesic with Random Curvature,” Vychisl. Metody Programm. 5, 291-296 (2004).
  5. M. E. Artyushkova and D. D. Sokoloff, “Numerical Modeling of the Solutions of the Jacobi Equation on a Geodesic with Random Curvature,” Astron. Zh. 82 (7), 584-589 (2005) [Astron. Rep. 49 (7), 520-525 (2005)].
  6. M. E. Artyushkova and D. D. Sokoloff, “Modelling Small-Scale Dynamo by the Jacobi Equation,” Magnetohydrodynamics 42 (1), 3-20 (2006).
  7. D. A. Grachev and D. D. Sokoloff, “Numerical Modeling of Growth of Multiplicative Random Quantities,” Vychisl. Metody Programm. 8, 1-5 (2007).
  8. V. G. Lamburt, D. D. Sokolov, and V. N. Tutubalin, “Jacobi Fields along a Geodesic with Random Curvature,” Mat. Zametki 74 (3), 416-424 (2003) [Math. Notes 74 (3), 393-400 (2003)].
  9. V. G. Lamburt, D. D. Sokolov, and V. N. Tutubalin, “Variety of the Variable Curvature and Asymptotic Results on the Production of Random Matrices,” in Proc. Int. Conf. on Mathematical Physics, Mathematical Simulation, and Approximate Methods, Obninsk, Russia, May 15-19, 2000 (Nuclear Power Engineering Inst., Obninsk, 2000), pp. 37-38.
  10. S. A. Molchanov, A. A. Ruzmaikin, and D. D. Sokolov, “Kinematic Dynamo in Random Flow,” Usp. Fiz. Nauk 145 (4), 593-628 (1985) [Sov. Phys. Usp. 28 (4), 307-327 (1985)].
  11. S. A. Molchanov, A. A. Ruzmaikin, and D. D. Sokolov, “Equations of Dynamo in Random Velocity Field with Short Correlation Time,” Magn. Gidrodyn. 4, 67-72 (1983) [Magnetohydrodynamics 19, 402-407 (1983)].
  12. N. Kleeorin, I. Rogachevskii, and D. Sokoloff, “Magnetic Fluctuations with Zero Mean Field in a Random Fluid with a Finite Correlation Time and a Small Magnetic Diffusion,” Phys. Rev. E 65 (3), 036303-036307 (2002).
  13. D. A. Grachev, “Memory Effects in the Problem of Light Propagation in a Universe with Inhomogeneities,” Vestn. Mosk. Univ., Ser. 3: Fiz., No. 1, 16-19 (2008) [Moscow Univ. Phys. Bull. 63 (1), 16-19 (2008)].
  14. D. A. Grachev and D. D. Sokoloff, “Higher Statistical Moments of the Solution to the Jacobi Equation with Random Curvature,” in Mathematical Models and Boundary Value Problems (Gos. Tekh. Univ., Samara, 2008), Part 3, pp. 83-86.
  15. D. A. Grachev, “Averaging of Jacobi Fields along Geodesics on Manifolds of Random Curvature,” J. Math. Sci. 160 (1), 128-138 (2009).
  16. D. A. Grachev, “Tensor Approach to the Problem of Averaging Differential Equations with δ-Correlated Random Coefficients,” Mat. Zametki 87 (3), 359-368 (2010) [Math. Notes 87 (3-4), 336-344 (2010)].
  17. E. A. Mikhailov, D. D. Sokoloff, and V. N. Tutubalin, “The Fundamental Matrix for the Jacobi Equation with Random Coefficients,” Vychisl. Metody Programm. 11, 261-268 (2010).
  18. E. A. Mikhailov and I. I. Modyaev, “Galactic Dynamo Equations with Random Coefficients,” Vychisl. Metody Programm. 15, 351-358 (2014).
  19. E. A. Mikhailov and V. V. Pushkarev, “Fluctuations of the Turbulent Diffusion Coefficient in Galaxy Dynamo Equations,” Vychisl. Metody Programm. 17, 447-454 (2016).



How to Cite

Грачев Д.А., Михайлов Е.А. Numerical Modeling of a Two-Point Correlator for the Lagrange Solutions of Some Evolution Equations // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2017. 18. 277-283. doi 10.26089/NumMet.v18r324



Section 1. Numerical methods and applications

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