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Analog of the monotone scheme to solve the non-self-conjugated set of quasi-diffusion equations in r–z geometry

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Abstract

The way to generate the monotonous difference scheme similar to non-self-conjugated quasi-diffusion equations in r–z geometry is examined through an example of a nonstationary problem of external isotropic radiation propagation in a cylindrical pipe. For this purpose, the coordinates in a plane (r, z) are rotated in such a way that the quasi-diffusion tensor takes a diagonal form in the center of the cell and, therefore, the nondiagonal elements on the side of the cell are minimized. This scheme is similar to the scheme that we have already developed for a self-conjugated problem [1]. The following hybrid difference scheme is used in calculations: it is non-monotone in the areas of smooth solutions and is analog to monotone at the contact boundaries. Note that for the plane case the non-monotone difference scheme is invariant relative to the rotation of the coordinates. Since the front of the light wave is normal to the contact boundary, the problem of the penetration of the external radiation in the pipe is useful in verifying the quality of the scheme

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References

  1. E. N. Aristova and A. V. Kolpakov, “Combined Difference Scheme for Approximation the Elliptic Operator on Non-Rectangular Cell,” Matematicheskoe modelirovanie 3(4), 93–102 (1991).

    MathSciNet  Google Scholar 

  2. J. A. Fleck, Jr. and J. D. Cummings, “An Implicit Monte Carlo Scheme for Calculating Time and Frequency Dependent Nonlinear Radiation Transport,” J. of Computational Physics 8(3), 313–342 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  3. M. L. Adams and E. W. Larsen, “Fast Iterative Methods for Discrete-Ordinates Particle Transport Calculations,” Progress in Nuclear Energy 40(1), 3–159 (2002).

    Article  Google Scholar 

  4. V. Ya. Gol’din, “Quasi-Diffuse Method for Solving the Kinetic Equation,” Zh. vychisl. matem. i matem. fiziki 4(6), 1078–1087 (1964).

    MathSciNet  Google Scholar 

  5. V. Ya. Gol’din, “On Mathematical Simulation of Continuous Medium Problems with Nonequilibrium Transport,” in Modern Problems of Mathematical Physics and Computational Mathematics (Nauka, Moscow, 1982) [in Russian].

    Google Scholar 

  6. E. N. Aristova and V. Ya. Gol’din, “The Method of Consideration of a Strong Scattering Anisotropy in Transport Equation,” in Proc. of the Joint Intern. Conf. on Mathematical Methods and Supercomputing for Nuclear Applications, Saratoga Springs, New York, Oct. 5–9 1997 (American Nuclear Society, Inc., La Grand Park, Illinois 60526 USA, 1997), vols. 1–2, pp. 1507–1516.

    Google Scholar 

  7. E. N. Aristova and V. Ya. Gol’din, “Computation of Anisotropy Scattering of Solar Radiation in Atmosphere (Monoenergetic Case),” Journal of Quantative Spectroscopy and Radiative Transfer 67, 139–157 (2000).

    Article  Google Scholar 

  8. V. Ya. Gol’din, V. A. Degtyarev, and A. V. Kolpakov, “Approximate Consideration of Nonstationarity in Quasi-Diffuse Equations,” Preprint No. 122 (Institute of Applied Mathematics, Moscow, 1984).

    Google Scholar 

  9. E. N. Aristova, D. F. Baidin, and V. Ya. Gol’din, “Two Variants of Economical Method to Solve the Transport Equation in r-z Geometry with the Help of Vladimirov Variables,” Matematicheskoe modelirovanie 18(7), 43–52 (2006).

    MATH  Google Scholar 

  10. E. N. Aristova, V. Ya. Gol’din, and A. V. Kolpakov, “Design Procedure of Radiation Transport in Solid of Revolution,” Matematicheskoe modelirovanie 9(3), 91–108 (1997).

    MATH  MathSciNet  Google Scholar 

  11. V. Ya. Gol’din and A. V. Kolpakov, “Nonlinear Method of Flow Run for Solving the Multidimentional Diffusion Equation,” Preprint No. 22 (Institute of Applied Mathematics, Moscow, 1982).

    Google Scholar 

  12. E. N. Aristova and V. Ya. Gol’din, “Nonlinear Speed-up of Iteration for Solving the Elliptic Set of Equations,” Matematicheskoe modelirovanie 13(9), 82–90 (2001).

    MATH  MathSciNet  Google Scholar 

  13. D. Yu. Anistratov, E. N. Aristova, and V. Ya. Gol’din, “Nonlinear Method to Solve the Problems of Radiation Transport in Medium,” Matematicheskoe modelirovanie 8(12), 3–29 (1996).

    MATH  MathSciNet  Google Scholar 

  14. V. Ya. Gol’din, D. A. Gol’dina, and A. V. Kolpakov, “On Solving 2D Stationary Problem of Quasi-Diffusion,” Preprint No. 49 (Institute of Applied Mathematics, Moscow, 1982).

    Google Scholar 

  15. E. N. Aristova, V. Ya. Gol’din, and A. V. Kolpakov, “Radiative Transport through Annular Slot in Solid of Revolution,” Matematicheskoe modelirovanie 9(4), 3–10 (1997).

    MATH  MathSciNet  Google Scholar 

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  1. Institute for Mathematical Modelling, Russian Academy of Sciences, pl. Miusskaya 4a, Moscow, 125047, Russia

    E. N. Aristova

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  1. E. N. Aristova
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Correspondence to E. N. Aristova.

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Original Russian Text © E.N. Aristova, 2009, published in Matematicheskoe Modelirovanie, 2009, Vol. 21, No. 2, pp. 47–59.

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Aristova, E.N. Analog of the monotone scheme to solve the non-self-conjugated set of quasi-diffusion equations in r–z geometry. Math Models Comput Simul 1, 745–756 (2009). https://doi.org/10.1134/S207004820906009X

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  • DOI: https://doi.org/10.1134/S207004820906009X

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