Error Estimates of Integral Deferred Correction Methods for Stiff Problems

In this paper, we present an error estimate of integral deferred correction (IDC) method constructed with stiffly accurate implicit Runge-Kutta (R-K) method for singular perturbation problems containing a stiff parameter \epsilon. We focus our analysis on the IDC method using uniform distribution of quadrature nodes, but excluding the left-most endpoint. The uniform distribution of nodes is important for high order accuracy increase in correction loops [5], where as the use of quadrature nodes excluding the left-most endpoint lead to an important stability condition for stiff problem, i.e. the method becomes L-stable if A-stable. In our error estimate, we expand the global error in powers of " and show convergence results for these error terms as well as the remainder. Specifically, the order of convergence for the first term in global error (index 1) increase with high order if a high order R-K method is applied in the IDC correction step; the order of convergence for the second term (index 2) is determined by the stage order of the R-K method for the IDC prediction. Numerical results for the stiff van der Pol equation are demonstrated to verify our error estimate.