Order results for implicit Runge-Kutta methods applied to differential/algebraic systems. (English) Zbl 0635.65084

This paper is concerned with the local and global behaviour of implicit Runge-Kutta methods when applied to both constant-coefficient linear and nonlinear (linear in y’) index one systems of differential algebraic equations (DAEs). It is shown that there are additional algebraic conditions that a Runge-Kutta method must satisfy, as well as the usual differential order conditions, and that in general there is an order reduction when Runge-Kutta methods are applied to DAEs. Furthermore, if in the nonlinear case, \(| 1-b^ TA^{-1}e| <1\) and the initial conditions satisfy \(\| y_ 0-y(t_ 0)\| =O(h^ G)\) where G is the minimum of the orders for the purely differential nonstiff part and the stage order \(+1\), then it is shown that the global error is at least \(O(h^ G)\). Some numerical experiments confirm the order reduction effects. (There is a slight error in the proof on p. 849 which does not, however, change the conclusions.)
Reviewer: K.Burrage


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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