The basic concepts and main features of Runge-Kutta and Rosenbrock exponential integrators will be reviewed, along with the main difficulties in computing the exponential matrix. The advantages of these methods for the numerical solution of large and stiff systems of nonlinear ordinary differential equations will be analyzed, on the basis of theorical results and numerical experiments. First examples of applications to test problems relevant for geophysical scale flows, climate modeling and numerical weather prediction will also be presented. The main open issues and perspectives for further developments will be evaluated and discussed.
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