  We present a new class of iterative schemes for solving initial value
problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the
weak DG formulation of an IVP, we derive a new iterative method based on a
preconditioned Picard iteration. Using this approach, we can systematically
construct explicit, implicit and semi-implicit schemes with arbitrary order of
accuracy. We also show that the same schemes can be constructed by solving a
series of correction equations based on the DG weak formulation. The accuracy
of the schemes is proven to be $\min\{2p+1, K+1\}$ with $p$ the degree of the
DG polynomial basis and $K$ the number of iterations. The stability is explored
numerically; we show that the implicit schemes are $A$-stable at least for $0
\leq p \leq 9$. Furthermore, we combine the methods with a multilevel strategy
to accelerate their convergence speed. The new multilevel scheme is intended to
provide a flexible framework for high order space-time discretizations and to
be coupled with space-time multigrid techniques for solving partial
differential equations (PDEs). We present numerical examples for ODEs and PDEs
to analyze the performance of the new methods. Moreover, the newly proposed
class of methods, due to its structure, is also a competitive and promising
candidate for parallel in time algorithms such as Parareal, PFASST, multigrid
in time, etc.
