  We consider quadrature formulas of high order in time based on Radau-type,
L-stable implicit Runge-Kutta schemes to solve time dependent stiff PDEs.
Instead of solving a large nonlinear system of equations, we develop a method
that performs iterative deferred corrections to compute the solution at the
collocation nodes of the quadrature formulas. The numerical stability is
guaranteed by a dedicated operator splitting technique that efficiently handles
the stiffness of the PDEs and provides initial and intermediate solutions to
the iterative scheme. In this way the low order approximations computed by a
tailored splitting solver of low algorithmic complexity are iteratively
corrected to obtain a high order solution based on a quadrature formula. The
mathematical analysis of the numerical errors and local order of the method is
carried out in a finite dimensional framework for a general semi-discrete
problem, and a time-stepping strategy is conceived to control numerical errors
related to the time integration. Numerical evidence confirms the theoretical
findings and assesses the performance of the method in the case of a stiff
reaction-diffusion equation.
