Implicit Runge-Kutta methods for transferable differential-algebraic equations

M. Arnold

Banach Center Publications (1994)

  • Volume: 29, Issue: 1, page 267-274
  • ISSN: 0137-6934

Abstract

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The numerical solution of transferable differential-algebraic equations (DAE’s) by implicit Runge-Kutta methods (IRK) is studied. If the matrix of coefficients of an IRK is non-singular then the arising systems of nonlinear equations are uniquely solvable. These methods are proved to be stable if an additional contractivity condition is satisfied. For transferable DAE’s with smooth solution we get convergence of order m i n ( k E , k I + 1 ) , where k E is the classical order of the IRK and k I is the stage order. For transferable DAE’s with generalized solution convergence of order 1 is ensured, provided that k E ≥ 1 .

How to cite

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Arnold, M.. "Implicit Runge-Kutta methods for transferable differential-algebraic equations." Banach Center Publications 29.1 (1994): 267-274. <http://eudml.org/doc/262717>.

@article{Arnold1994,
abstract = {The numerical solution of transferable differential-algebraic equations (DAE’s) by implicit Runge-Kutta methods (IRK) is studied. If the matrix of coefficients of an IRK is non-singular then the arising systems of nonlinear equations are uniquely solvable. These methods are proved to be stable if an additional contractivity condition is satisfied. For transferable DAE’s with smooth solution we get convergence of order $min(k_E,k_I + 1)$, where $k_E$ is the classical order of the IRK and $k_I$ is the stage order. For transferable DAE’s with generalized solution convergence of order 1 is ensured, provided that $k_E ≥ 1$.},
author = {Arnold, M.},
journal = {Banach Center Publications},
keywords = {convergence; differential-algebraic equations; implicit Runge-Kutta methods; stability function},
language = {eng},
number = {1},
pages = {267-274},
title = {Implicit Runge-Kutta methods for transferable differential-algebraic equations},
url = {http://eudml.org/doc/262717},
volume = {29},
year = {1994},
}

TY - JOUR
AU - Arnold, M.
TI - Implicit Runge-Kutta methods for transferable differential-algebraic equations
JO - Banach Center Publications
PY - 1994
VL - 29
IS - 1
SP - 267
EP - 274
AB - The numerical solution of transferable differential-algebraic equations (DAE’s) by implicit Runge-Kutta methods (IRK) is studied. If the matrix of coefficients of an IRK is non-singular then the arising systems of nonlinear equations are uniquely solvable. These methods are proved to be stable if an additional contractivity condition is satisfied. For transferable DAE’s with smooth solution we get convergence of order $min(k_E,k_I + 1)$, where $k_E$ is the classical order of the IRK and $k_I$ is the stage order. For transferable DAE’s with generalized solution convergence of order 1 is ensured, provided that $k_E ≥ 1$.
LA - eng
KW - convergence; differential-algebraic equations; implicit Runge-Kutta methods; stability function
UR - http://eudml.org/doc/262717
ER -

References

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  1. [Arn90] M. Arnold, Numerische Behandlung von semi-expliziten Algebrodifferentialgleichungen vom Index 1 mit linear-impliziten Verfahren, PhD thesis, Martin-Luther-Universität Halle, Sektion Mathematik, 1990. 
  2. [But87] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, Wiley, Chichester 1987. 
  3. [DHZ87] P. Deuflhard, E. Hairer and J. Zugck, One-step and extrapolation methods for differential-algebraic systems, Numer. Math. 51 (1987), 501-516. Zbl0635.65083
  4. [GM86] E. Griepentrog and R. März, Differential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte Math. 88, Leipzig 1986. 
  5. [HLR89] E. Hairer, Ch. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Math. 1409, Springer, Berlin 1989. 
  6. [Ob90] H. Oberdörfer, Zur numerischen Behandlung von Algebrodifferentialgleichungen mit Runge-Kutta-Verfahren, PhD thesis, Ernst-Moritz-Arndt-Universität Greifswald, Sektion Mathematik, 1990. 
  7. [ORh70] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York 1970. Zbl0241.65046
  8. [Pe86] L. R. Petzold, Order results for implicit Runge-Kutta methods applied to differential/algebraic systems, SIAM J. Numer. Anal. 23 (1986), 837-852. Zbl0635.65084

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