The Approximate Runge-Kutta Computational Process

Implicit Runge-Kutta methods are efficient for solving stiff ODEs and DAEs. To bridge the gap between their theoretical analysis and practical implementation, we introduce the notion of the (Δ, K)-approximate Runge- Kutta process to account for inevitable iteration errors. We prove iteration error bounds uniform with respect to stiffness, and investigate stage derivative reuse for methods having a first explicit stage. The latter technique may result in significant performance gains, also when such methods are used as error estimators. Previous computational heuristics can therefore be replaced by a consistent approach supported by theoretical analysis. The approximate but well-defined computational process is evaluated using approved test problems.