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Monotone Schemes of Conditional Approximation and Arbitrary Order of Accuracy for the Transport Equation

Abstract

An initial-boundary value problem for the one-dimensional transport equation with a constant coefficient \(a > 0\) is approximated by a usual explicit explicit monotone difference scheme of traveling calculation “upwind scheme”. Under a Courant-type condition, it is proved that the scheme has an arbitrary \(k\)th order of accuracy for smooth solutions. Assuming the existence of weakly discontinuous solutions, the results are generalized to multidimensional equations. Monotone finite difference schemes for equations with variable coefficients and for first-order semilinear hyperbolic equations are constructed with the use of a special Steklov averaging with respect to nonlinearity. The efficiency of the considered methods is illustrated by numerical results.

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REFERENCES

  1. N. N. Kalitkin and I. A. Kozlitin, “Comparison of difference schemes for the transfer equation,” Mat. Mod. 18 (4), 35–42 (2006).

    MATH  Google Scholar 

  2. P. J. Roache, Computational Fluid Dynamics (Hermosa, Albuquerque, 1976).

    MATH  Google Scholar 

  3. S. K. Godunov, “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,” Mat. Sb. 47, 271–306 (1959).

    MathSciNet  MATH  Google Scholar 

  4. A. I. Tolstykh, “On families of compact fourth- and fifth-order approximations involving the inversion of two-point operators for equations with convective terms,” Comput. Math. Math. Phys. 50 (5), 848–861 (2010).

    MathSciNet  Article  Google Scholar 

  5. A. I. Tolstykh, Compact Finite Difference Schemes and Application in Aerodynamic Problems (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  6. T. A. Aleksandrikova, M. P. Galanin, and T. G. Elenina, “Nonlinear monotonization of K.I. Babenko scheme for the numerical solution of the advection equation,” Mat. Mod. 16 (6), 44–47 (2004).

    MATH  Google Scholar 

  7. V. M. Goloviznin and S. A. Karabasov, “Balancing characteristic schemes on piecewise constant initial data: Jump transfer,” Mat. Model. 15 (10), 71–83 (2003).

    MathSciNet  MATH  Google Scholar 

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

  9. M. P. Galanin, “A nonlinear quasi-monotone finite element scheme for solving two-dimensional problems with transfer: A problem on a velocity skin layer,” Differ. Equations 32 (7), 941–949 (1996).

    MathSciNet  MATH  Google Scholar 

  10. P. N. Vabishchevich, V. A. Pervichko, A. A. Samarskii, and V. V. Chudanov, “Nonlinear regularized finite-difference schemes for the multidimensional transport equation,” Comput. Math. Math. Phys. 40 (6), 860–867 (2000).

    MathSciNet  MATH  Google Scholar 

  11. K. V. Vyaznikov, V. F. Tishkin, and A. P. Favorskii, “Construction of high-order accurate monotone difference schemes for systems of hyperbolic equations,” Mat. Model. 1 (5), 95–120 (1989).

    MathSciNet  MATH  Google Scholar 

  12. K. V. Vyaznikov, “High-order accurate quasi-monotone difference schemes on nonuniform meshes,” Mat. Model. 2 (3), 127–149 (1990).

    MathSciNet  MATH  Google Scholar 

  13. S. V. Ershov, “High-order accurate quasi-monotone ENO scheme for integrating the Euler and Navier–Stokes equations,” Mat. Model. 6 (11), 63–75 (1994).

    MathSciNet  MATH  Google Scholar 

  14. P. N. Vabishchevich, “Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics,” Comput. Math. Math. Phys. 50 (1), 112–123 (2010).

    MathSciNet  Article  Google Scholar 

  15. P. N. Vabishchevich, “Two-level schemes of higher approximation order for time-dependent problems with skew-symmetric operators,” Comput. Math. Math. Phys. 51 (6), 1050–1060 (2011).

    MathSciNet  Article  Google Scholar 

  16. N. Ya. Moiseev and I. Yu. Silant’eva, “Arbitrary-order difference schemes for solving linear advection equations with constant coefficients by the Godunov method with antidiffusion,” Comput. Math. Math. Phys. 48 (7), 1210–1220 (2008).

    MathSciNet  Article  Google Scholar 

  17. V. I. Paasonen, “Properties of difference schemes with oblique stencils for hyperbolic equations,” Numer. Anal. Appl. 11 (1), 60–72 (2018).

    MathSciNet  Article  Google Scholar 

  18. B. J. 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).

    Article  Google Scholar 

  19. P. Colella and P. R. Woodward, “The piecewise parabolic method (PPM) for gas-dynamical simulations,” J. Comput. Phys. 54 (1), 174–201 (1984).

    Article  Google Scholar 

  20. A. Harten, “High resolutions schemes for hyperbolic conservation laws,” J. Comput. Phys. 49, 357–393 (1983).

    MathSciNet  Article  Google Scholar 

  21. V. M. Goloviznin, S. A. Karabasov, and I. M. Kobrinskii, “Balance-characteristic schemes with separated conservative and flux variables”, Mat. Model. 15 (9), 29–48 (2003).

    MathSciNet  MATH  Google Scholar 

  22. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

  23. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed. (Wiley, New York, 1967).

    MATH  Google Scholar 

  24. A. Samarskii, P. Vabishchevich, and P. Matus, Difference Schemes with Operator Factors (Kluwer Academic, London, 2002).

    Book  Google Scholar 

  25. P. Matus, V. Irkhin, and M. Łapińska-Chrzczonowicz, “Exact difference schemes for time-dependent problems,” Comput. Methods Appl. Math. 5 (4), 422–448 (2005).

    MathSciNet  Article  Google Scholar 

  26. P. P. Matus, V. A. Irkhin, M. Łapińska-Chrzczonowicz, and S. V. Lemeshevsky, “On exact finite-difference schemes for hyperbolic and elliptic equations,” Differ. Equations 43 (7), 1001–1010 (2007).

    MathSciNet  Article  Google Scholar 

  27. S. Lemeshevsky, P. Matus, and D. Poliakov, Exact Finite-Difference Schemes (Walter de Gruyter, Berlin, 2016).

    Book  Google Scholar 

  28. P. Matus, “The maximum principle and some of its applications,” Comput. Methods Appl. Math. 2 (1), 50–91 (2002).

    MathSciNet  Article  Google Scholar 

  29. N. N. Kalitkin, “The Euler–McLaren formulas of high orders”, Mat. Model. 16 (10), 64–66 (2004).

    MATH  Google Scholar 

  30. Tingchun Wang, “Convergence of an eight-order compact difference scheme for the nonlinear Schrödinger equation,” Adv. Numer. Anal. 2012, 1–24 (2012).

    Article  Google Scholar 

  31. S. A. II’in and E. V. Timofeev, “A comparison of quasi-monotone finite difference shock-capturing schemes on the basis of the Cauchy problem for one-dimensional linear transport equation,” Mat. Model. 4 (3), 62–75 (1992).

    MathSciNet  Google Scholar 

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ACKNOWLEDGMENTS

The authors are grateful to M.P. Galanin for helpful discussions concerning the subject of this paper.

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Correspondence to P. P. Matus or B. D. Utebaev.

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Translated by I. Ruzanova

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Matus, P.P., Utebaev, B.D. Monotone Schemes of Conditional Approximation and Arbitrary Order of Accuracy for the Transport Equation. Comput. Math. and Math. Phys. 62, 359–371 (2022). https://doi.org/10.1134/S0965542522030101

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  • DOI: https://doi.org/10.1134/S0965542522030101

Keywords:

  • monotone scheme
  • exact difference scheme
  • schemes of arbitrary order
  • Steklov averaging
  • conditional approximation