Error estimates for deferred correction methods in time

We consider the deferred correction principle for high order accurate time discretization of ordinary differential equations (ODEs) and partial differential equations (PDEs) using the method of lines. We derive error estimates for the deferred correction method based on the implicit midpoint rule and the implicit BDF2 scheme and it is shown that the methods have no stability restraint on the size of the timestep. We are interested in the application of deferred correction schemes to initial boundary value problems. The treatment of the boundary conditions has shown to be nontrivial for many high order time discretization methods. Straightforward implementation of the boundary conditions leads to a reduction of the order of accuracy in time. We present a way to treat the boundary conditions to retain high order approximations.