Advertisement

Convergence of Alternant Theta-Method with Applications

  • István FaragóEmail author
  • Zénó Farkas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)

Abstract

In this work the alternant theta-method and its application is investigated. We analyze the local approximation error and the convergence of the method on the non-equidistant mesh. We define the order of convergence, as well. The main idea of this approach is the approximation of the solution of the Cauchy problems by using different numerical schemes (implicit, explicit, IMEX, one-step, multi-step etc.) with varying step-sizes. Benefits of such approximations are shown for the problems with non-smooth solutions. We show that the convergence and the error estimation can be given relatively easily for the classical \(\theta \)-method in case both equidistant and non-equidistant time/space discretizations. We analyze the connection of this approach to the classical discrete Gronwall lemma. We show that the extended discrete Gronwall lemma can be successfully applied to the estimation of the convergence’s rate of the alternant \(\theta _i\) method. We show numerical examples for some non-linear time dependent differential equations, which have non-continuous or strongly oscillated solutions.

Keywords

Alternant theta method Stability constant Gronwall lemma 

References

  1. 1.
    Faragó, I.: Convergence and stability constant of the theta-method. Appl. Math. 58, 42–51 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Emmrich, E.: Discrete versions of Gronwall’s lemma and their application to the numerical analysis of parabolic problems. Fachbereich 3 Preprint Reihe Mathematik 637, 10–13 (1999). Berlin TUGoogle Scholar
  3. 3.
    Clark, D.S.: Short proof of a discrete Gronwall inequality. Discrete Appl. Math. 16(3), 279–281 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Holte, J.M.: Discrete Gronwall lemma. In: MAA-NCS Meeting at the University of North Dakota (2009)Google Scholar
  5. 5.
    Pachpatte, B.G., Singare, S.M.: Discrete generalized Gronwall inequalities in three independent variables. Pacific J. Math. 82(1), 197–210 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Wiley, New York (1966)zbMATHGoogle Scholar
  7. 7.
    Suli, E.: Numerical Solution of Ordinary Differential Equations. Cambridge University Press, Oxford (2010)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Applied Analysis and Computational MathematicsEötvös Loránd UniversityBudapestHungary
  2. 2.MTA-ELTE NumNet Research GroupBudapestHungary

Personalised recommendations