Numerical integration of large stiff nonlinear systems of ODEs over
long time intervals is a challenging task since the stiffness of the
system makes solving them using standard explicit and implicit
methods computationally expensive. Exponential propagation-type
techniques offer significant advantages for these problems compared
to standard integrators. We will discuss new exponential propagation
iterative (EPI) methods, which are constructed by approximating the
integral form of the solution to a nonlinear autonomous system of
ODEs by an expansion in terms of products between special functions
of matrices and vectors. The matrix functions-vector products are
then approximated using Krylov subspace projections. For problems
where no good preconditioner is available, the EPI integrators can
outperform standard methods since they possess superior stability
properties compared to explicit schemes and offer computational
savings compared to implicit Newton-Krylov integrators by requiring
fewer Arnold iterations per time step. As an application, for which
exponential propagation methods offer computational savings, I will
discuss modeling large-scale plasma behavior using resistive MHD
equations. I will present results of a numerical model which
describes plasma as a time-dependent driven system. The results of
the simulations suggest new structure of plasma configurations that
form in the course of evolution of solar coronal arcades or
laboratory spheromak-type plasmas.