  Exponential integrators are special time discretization methods where the
traditional linear system solves used by implicit schemes are replaced with
computing the action of matrix exponential-like functions on a vector. A very
general formulation of exponential integrators is offered by the Exponential
Propagation Iterative methods of Runge-Kutta type (EPIRK) family of schemes.
The use of Jacobian approximations is an important strategy to drastically
reduce the overall computational costs of implicit schemes while maintaining
the quality of their solutions. This paper extends the EPIRK class to allow the
use of inexact Jacobians as arguments of the matrix exponential-like functions.
Specifically, we develop two new families of methods: EPIRK-W integrators that
can accommodate any approximation of the Jacobian, and EPIRK-K integrators that
rely on a specific Krylov-subspace projection of the exact Jacobian. Classical
order conditions theories are constructed for these families. A practical
EPIRK-W method of order three and an EPIRK-K method of order four are
developed. Numerical experiments indicate that the methods proposed herein are
computationally favorable when compared to existing exponential integrators.
