Advertisement

Time Step for Numerically Solving Parabolic Problems

  • Petr N. VabishchevichEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

Abstract

This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via explicit calculations. Using the explicit scheme, we calculate the solution at a new time level. We employ this solution in order to obtain the solution at the previous time level (the implicit scheme, explicit calculations). This solution should be close to the solution of our problem at this time level with a prescribed accuracy. Such an algorithm leads to explicit formulas for the calculation of the time step and takes into account both the dynamics of the problem solution and changes in coefficients of the equation and in its right-hand side.

Notes

Acknowledgements

This work was supported by RFBR (project 14-01-00785).

References

  1. 1.
    Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (1998)Google Scholar
  2. 2.
    Gear, C.W.: Numerical initial value problems in ordinary differential equations. Prentice Hall (1971)Google Scholar
  3. 3.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Heidelberg (1987)zbMATHCrossRefGoogle Scholar
  4. 4.
    Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)zbMATHCrossRefGoogle Scholar
  5. 5.
    LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-state and Time-dependent Problems. Society for Industrial and Applied Mathematics, Philadelphia (2007)CrossRefGoogle Scholar
  6. 6.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Heidelberg (2010)Google Scholar
  7. 7.
    Vabishchevich, P.N.: A priori estimation of a time step for numerically solving parabolic problems. Appl. Math. Comput. 250, 424–431 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Nuclear Safety InstituteMoscowRussia
  2. 2.North-Eastern Federal UniversityYakutskRussia

Personalised recommendations