Mathematical Models and Computer Simulations

, Volume 10, Issue 1, pp 79–88 | Cite as

A Fourth-Order Accurate Difference Scheme for a Differential Equation with Variable Coefficients

  • V. A. GordinEmail author
  • E. A. Tsymbalov


A compact difference scheme on a three-point stencil for an unknown function is proposed. The scheme approximates a second-order linear differential equation with a variable smooth coefficient. Our numerical experiment confirms the fourth order of accuracy of the solution of the difference scheme and of the approximation of the eigenvalues of the boundary problem. The difference operator is almost self-adjoint and its spectrum is real. Richardson extrapolation helps to increase the order of accuracy.


compact difference scheme divergent scheme test functions self-conjugacy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. A. Gordin, How Should You Calculate It? (MCCME, Moscow, 2005) [in Russian].Google Scholar
  2. 2.
    V. A. Gordin, Mathematics, Computer, Weather Forecasting and Other Scenarios of Mathematical Physics (Fizmatlit, Moscow, 2010, 2013) [in Russian].Google Scholar
  3. 3.
    A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1966; Dover, New York, 1990).zbMATHGoogle Scholar
  4. 4.
    A. A. Samarskii, Introduction to the Difference Schemes (Nauka, Moscow, 1971; CRC, Boca Raton, FL, 2001).CrossRefzbMATHGoogle Scholar
  5. 5.
    R. P. Fedorenko, Lectures on Computational Physics (MFTI, Moscow, 1994; Intellekt, Dolgoprudnyi, 2008) [in Russian].Google Scholar
  6. 6.
    M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Nauka, Moscow, 1983; Springer Science, New York, 2012).Google Scholar
  7. 7.
    A. A. Abramov, “On boundary conditions at singular point for system of ordinary differential equations,” Zh. Vychisl. Mat. Mat. Fiz. 11, 275–278 (1971).MathSciNetzbMATHGoogle Scholar
  8. 8.
    V. A. Gordin and E. A. Tsymbalov, “Compact differential schemes for the diffusion and Schrödinger equations. Approximation, stability, convergence, effectiveness, monotony,” J. Comput. Math. 32, 348–370 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. A. Gordin and E. A. Tsymbalov, “Compact difference scheme for the differential equation with piecewiseconstant coefficient,” Mat. Model. 27 (12), 16–28 (2017).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Higher School of EconomicsNational Research UniversityMoscowRussia
  2. 2.Hydrometeorological Center of RussiaMoscowRussia
  3. 3.Skolkovo Institute of Science and TechnologyMoscowRussia

Personalised recommendations