Advertisement

BIT Numerical Mathematics

, Volume 52, Issue 2, pp 437–455 | Cite as

A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise

  • Dominique Küpper
  • Anne Kværnø
  • Andreas RößlerEmail author
Article

Abstract

The paper deals with the numerical treatment of stochastic differential-algebraic equations of index one with a scalar driving Wiener process. Therefore, a particularly customized stochastic Runge-Kutta method is introduced. Order conditions for convergence with order 1.0 in the mean-square sense are calculated and coefficients for some schemes are presented. The proposed schemes are stiffly accurate and applicable to nonlinear stochastic differential-algebraic equations. As an advantage they do not require the calculation of any pseudo-inverses or projectors. Further, the mean-square stability of the proposed schemes is analyzed and simulation results are presented bringing out their good performance.

Keywords

Stochastic differential-algebraic equation Stochastic Runge-Kutta method Stiffly accurate Mean-square convergence Mean-square stability 

Mathematics Subject Classification (2000)

65C30 65L80 65L06 65L20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexander, R.: Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s. SIAM J. Numer. Anal. 14(6), 1006–1021 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics, vol. 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996). Revised and corrected reprint of the 1989 original zbMATHGoogle Scholar
  3. 3.
    Burrage, K., Tian, T.: Implicit stochastic Runge-Kutta methods for stochastic differential equations. BIT Numer. Math. 44(1), 21–39 (2004). doi: 10.1023/B:BITN.0000025089.50729.0f MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Debrabant, K., Kværnø, A.: B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47(1), 181–203 (2008/09). doi: 10.1137/070704307 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Debrabant, K., Rößler, A.: Diagonally drift-implicit Runge-Kutta methods of weak order one and two for Itô SDEs and stability analysis. Appl. Numer. Math. 59(3–4), 595–607 (2009). doi: 10.1016/j.apnum.2008.03.011 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Berlin (1989) zbMATHGoogle Scholar
  7. 7.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II, 2nd edn. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (2002). Stiff and differential-algebraic problems Google Scholar
  8. 8.
    Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (electronic) (2000). doi: 10.1137/S003614299834736X. MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991) zbMATHCrossRefGoogle Scholar
  10. 10.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992) zbMATHGoogle Scholar
  11. 11.
    Küpper, D.: Runge-Kutta methods for stochastic differential-algebraic equations. Ph.D. thesis, TU Darmstadt, Fachbereich Mathematik, Verlag Dr. Hut, München (2009) Google Scholar
  12. 12.
    Römisch, W., Winkler, R.: Stochastic DAEs in circuit simulation. In: Modeling, Simulation, and Optimization of Integrated Circuits (Oberwolfach, 2001). Internat. Ser. Numer. Math., vol. 146, pp. 303–318. Birkhäuser, Basel (2003) CrossRefGoogle Scholar
  13. 13.
    Rößler, A.: Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. Numer. Anal. 48(3), 922–952 (2010). doi: 10.1137/09076636X MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rößler, A.: Stochastic Taylor expansions for functionals of diffusion processes. Stoch. Anal. Appl. 28(3), 415–429 (2010). doi: 10.1080/07362991003707905 MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Saito, Y., Mitsui, T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33(6), 2254–2267 (1996). doi: 10.1137/S0036142992228409 MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Schein, O.: Stochastic differential-algebraic equations in circuit simulation. Ph.D. thesis, TU Darmstadt, Fachbereich Mathematik, Shaker Verlag, Aachen (1999) Google Scholar
  17. 17.
    Schein, O., Denk, G.: Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits. J. Comput. Appl. Math. 100(1), 77–92 (1998). doi: 10.1016/S0377-0427(98)00138-1 MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Sickenberger, T., Weinmüller, E., Winkler, R.: Local error estimates for moderately smooth problems. II. SDEs and SDAEs with small noise. BIT Numer. Math. 49(1), 217–245 (2009). doi: 10.1007/s10543-009-0209-0 zbMATHCrossRefGoogle Scholar
  19. 19.
    Winkler, R.: Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math. 157(2), 477–505 (2003). doi: 10.1016/S0377-0427(03)00436-9 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  • Dominique Küpper
    • 1
  • Anne Kværnø
    • 2
  • Andreas Rößler
    • 3
    Email author
  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  3. 3.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany

Personalised recommendations