On the construction of error estimators for implicit Runge-Kutta methods

For implicit Runge--Kutta methods intended for stiff ODEs or DAEs, it is often difficult to embed a local error estimating method which gives realistic error estimates for stiff/algebraic components. If the embedded method's stability function is unbounded at $z=infty$, stiff error components are grossly overestimated. In practice some codes ``improve'' such inadequate error estimates by premultiplying the estimate by a ``filter'' matrix which damps or removes the large, stiff error components. Although improving computational performance, this technique is somewhat arbitrary and lacks a sound theoretical backing. In this scientific note we resolve this problem by introducing an implicit error estimator. It has the desired properties for stiff/algebraic components without invoking artificial improvements. The error estimator contains a free parameter which determines the magnitude of the error, and we show how this parameter is to be selected on the basis of method properties. The construction principles for the error estimator can be adapted to all implicit Runge--Kutta methods, and a better agreement between actual and estimated errors is achieved, resulting in better performance.