  In this article, we employ the construction of the time-marching
Discontinuous Petrov-Galerkin (DPG) scheme we developed for linear problems to
derive high-order multistage DPG methods for non-linear systems of ordinary
differential equations. The methodology extends to abstract evolution equations
in Banach spaces, including a class of nonlinear partial differential
equations. We present three nested multistage methods: the hybrid Euler method
and the two- and three-stage DPG methods. We employ a linearization of the
problem as in exponential Rosenbrock methods, so we need to compute exponential
actions of the Jacobian that change from time steps. The key point of our
construction is that one of the stages can be post-processed from another
without an extra exponential step. Therefore, the class of methods we introduce
is computationally cheaper than the classical exponential Rosenbrock methods.
We provide a full convergence proof to show that the methods are second, third,
and fourth-order accurate, respectively. We test the convergence in time of our
methods on a 2D + time semi-linear partial differential equation after a
semidiscretization in space.
