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Exponential difference schemes for solving boundary-value problems for convection-diffusion type equations

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Abstract

The paper considers the numerical solution of boundary-value problems for multidimensional convection-diffusion type equations (CDEs). Such equations are useful for various physical processes in solids, liquids and gases. A new approach to the spatial approximation for such equations is proposed. This approach is based on an integral transformation of second-order one-dimensional differential operators. A linear version of CDE was chosen for simplicity of the analysis. In this setting, exponential difference schemes were constructed, algorithms for their implementation were developed, a brief analysis of the stability and convergence was made. This approach was numerically tested for a two-dimensional problem of motion of metallic particles in water flow subject to a constant magnetic field.

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References

  1. A. A. Samarskii and A. P. Mikhailov, Principles of Mathematical Modelling: Ideas, Methods, Examples (Fizmatlit, Moscow, 1997; Tailor Francis, London, (2002).

    MATH  Google Scholar 

  2. A. A. Samarskii and V. B. Andreev, Difference Methods for Elliptic Equations (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  3. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977; Marcel Dekker, New York, 2001).

    Book  MATH  Google Scholar 

  4. R. Li, Zh. Chen, and W. Wu, Generalized Difference Methods for Differential Equations. Numerical Analysis of Finite Volume Methods (Marcel Dekker, New York, (2000).

    MATH  Google Scholar 

  5. R. Eymard, T. R. Gallouet, and R. Herbin, “The finite volume method,” in Handbook of Numerical Analysis, Ed. by P. G. Ciarlet and J. L. Lions (North-Holland, Amsterdam, (2000), Vol. 7, pp. 713–1020.

    MATH  Google Scholar 

  6. R. J. LeVeque, Numerical Methods for Conservation Laws (Birkhauser, Basel, (2008).

    MATH  Google Scholar 

  7. M. P. Galanin and E. B. Savenkov, Methods for Numerical Analysis of Mathematical Models (Mosk. Gos. Tekh. Univ. im. Baumana, Moscow, (2010) [in Russian].

    MATH  Google Scholar 

  8. W. Gautschi, Numerical Analysis (Springer, Berlin, (2012).

    Book  MATH  Google Scholar 

  9. N. V. Karetkina, “An unconditionally stable difference scheme for parabolic equations containing first derivatives,” USSR Comput. Math. Math. Phys. 20, 257–262 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Doolan, J. Miller, and W. Shilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers (Boole, Dublin, (1980).

    MATH  Google Scholar 

  11. S. V. Polyakov and V. A. Sablikov, “Light-induced charge carriers lateral transfer in heterostructures with 2D electron gas,” Mat. Model. 9 (12), 76–86 (1997).

    MATH  Google Scholar 

  12. S. V. Polyakov, “Exponential schemes for solving evolutionary equations on non-regular grids,” Uch. Zap. Kazan. Univ., Ser.: Fiz.-Mat. Nauki 149 (4), 121–131 (2007).

    Google Scholar 

  13. Yu. N. Karamzin and S. V. Polyakov, “Exponential finite volume schemes for solving elliptic and parabolic general form equations on non-regular grids,” in Proceedings of the 8th All-Russia Conference on Grid Methods for Boundary Problems and Applications (Kazan. Gos. Univ., Kazan, (2010), pp. 234–248.

    Google Scholar 

  14. A. A. Samarskii, “Monotonic difference schemes for elliptic and parabolic equations in the case of a non-selfadjoint elliptic operator,” USSR Comput. Math. Math. Phys. 5, 212–217 (1965).

    Article  MATH  Google Scholar 

  15. E. I. Golant, “Conjugate families of difference schemes for equations of parabolic type with lowest terms,” USSR Comput. Math. Math. Phys. 18 (5), 88–95 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. V. Polyakov, “Exponential difference schemes for convection-diffusion equation,” Math. Montisnigri 25, 1–16 (2012).

    Google Scholar 

  17. S. V. Polyakov, “Exponential Difference Schemes with Double Integral Transformation for Solving Convection-Diffusion Equations,” Math. Models Comput. Simul. 5, 338–340 (2013).

    Article  Google Scholar 

  18. A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, Vol. 1: Direct Methods, Vol. 2: Iterative Methods (Birkhäuser, Basel, Boston, Berlin, 1989; Nauka, Moscow, (1976).

    Google Scholar 

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047, Russia

    S. V. Polyakov, Yu. N. Karamzin & T. A. Kudryashova

  2. Moscow Institute of Engineering Physics, Moscow, 115409, Russia

    S. V. Polyakov

  3. Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, 141700, Russia

    I. V. Tsybulin

Authors
  1. S. V. Polyakov
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  2. Yu. N. Karamzin
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  3. T. A. Kudryashova
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  4. I. V. Tsybulin
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Correspondence to S. V. Polyakov.

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Original Russian Text © S.V. Polyakov, Yu.N. Karamzin, T.A. Kudryashova, I.V. Tsybulin, 2016, published in Matematicheskoe Modelirovanie, 2016, Vol. 28, No. 7, pp. 121–136.

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Polyakov, S.V., Karamzin, Y.N., Kudryashova, T.A. et al. Exponential difference schemes for solving boundary-value problems for convection-diffusion type equations. Math Models Comput Simul 9, 71–82 (2017). https://doi.org/10.1134/S2070048217010124

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  • DOI: https://doi.org/10.1134/S2070048217010124

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