Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations

Abstract

Iterated Defect Correction (IDeC)-methods based on the implicit Euler scheme are shown to have a fixed point. This fixed point coincides with the solution of certain implicit multi-stage Runge-Kutta methods (equivalent to polynomial collocation). Sufficient conditions for the convergence of the iterates to the fixed point are given for linear problems. These results indicate that for a large variety of general non-linear stiff problems, fixed-point-convergence can be expected, and moreover they indicate that the rate of convergence to the fixed point is very high for very stiff problems. Thus the proposed methods combine the high orders and the high accuracy of multistage-methods with the low computational effort of single-stage methods.