Abstract :
In this paper we consider the construction, analysis, implementation and application
of exponential integrators. The focus will be on two types of stiff
problems. The first one is characterized by a Jacobian that possesses eigenvalues
with large negative real parts. Parabolic partial differential equations
and their spatial discretization are typical examples. The second class consists
of highly oscillatory problems with purely imaginary eigenvalues of large
modulus. Apart from motivating the construction of exponential integrators
for various classes of problems, our main intention in this article is to present
the mathematics behind these methods. We will derive error bounds that are
independent of stiffness or highest frequencies in the system.
Since the implementation of exponential integrators requires the evaluation
of the product of a matrix function with a vector, we will briefly discuss some
possible approaches as well. The paper concludes with some applications, in
which exponential integrators are used
