The numerical solution of higher index differential/algebraic equations by implicit methods

This paper studies the order, stability, and convergence properties of implicit Runge–Kutta (IRK) methods applied to differential/algebraic systems with index greater than one. These methods do not in general attain the same order of accuracy for higher index differential/algebraic systems as they do for index 1 systems or for purely differential systems. Necessary and sufficient conditions on the method coefficients are derived to ensure that the local and global errors of the method attain a given order of accuracy for higher index linear constant coefficient systems. IRK methods applied to nonlinear semi-explicit index 2 systems are studied, and a sufficient set of conditions is derived which ensures that a method is accurate to a given order for these systems. Finally, some numerical experiments are presented that illustrate the theoretical results and demonstrate the effects of roundoff' errors on the solution.

[1]  Per Lötstedt,et al.  Numerical solution of nonlinear differential equations with algebraic constraints I: Convergence results for backward differentiation formulas , 1986 .

[2]  A. Prothero,et al.  On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations , 1974 .

[3]  K. Brenan,et al.  Numerical simulation of trajectory prescribed path control problems by the backward differentiation formulas , 1986 .

[4]  K. Brenan,et al.  Difference approximations for higher index differential-algebraic systems with applications in trajectory control , 1984, The 23rd IEEE Conference on Decision and Control.

[5]  Christoph W. Ueberhuber,et al.  Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems , 1985 .

[6]  R. Alexander.  Diagonally implicit runge-kutta methods for stiff odes , 1977 .

[7]  V. Dolezal.  The existence of a continuous basis of a certain linear subspace of $E\sb{r}$ which depends on a parameter , 1964 .

[8]  C. W. Gear,et al.  Automatic integration of Euler-Lagrange equations with constraints , 1985 .

[9]  Kevin Burrage,et al.  A special family of Runge-Kutta methods for solving stiff differential equations , 1978 .

[10]  J. R. Cash,et al.  Diagonally Implicit Runge-Kutta Formulae with Error Estimates , 1979 .

[11]  L. Petzold.  A description of dassl: a differential/algebraic system solver , 1982 .

[12]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[13]  L. Petzold.  Order results for implicit Runge-Kutta methods applied to differential/algebraic systems , 1986 .

[14]  Kevin Burrage,et al.  Efficiently Implementable Algebraically Stable Runge–Kutta Methods , 1982 .

[15]  Syvert P. Nørsett,et al.  Runge-Kutta methods with a multiple real eigenvalue only , 1976 .