Error Estimates of Integral Deferred Correction Methods for Stiff Problems

Abstract

In this paper, we present error estimates of the integral deferred correction method constructed with stiffly accurate implicit Runge-Kutta methods with a nonsingular matrix $A$ in its Butcher table representation, when applied to stiff problems characterized by a small positive parameter $\varepsilon$. In our error estimates, we expand the global error in powers of $\varepsilon$ and show that the coefficients are global errors of the integral deferred correction method applied to a sequence of differential algebraic systems. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical results for the van der Pol equation are presented {to} illustrate our theoretical findings. Finally, we study the linear stability properties of these methods.