Exponential integrators

Numerical schemes for ordinary differential equations, using matrix exponentials, were introduced in the 1960's as a way to overcome the stability restrictions of explicit methods. However, such methods were not considered as a practical mean of solving stiff ODEs until quite recently. Due to improvements in the efficient computation of the exponential function, exponential integrators have emerged as a viable alternative for the integration of spatially discretized nonlinear parabolic and hyperbolic differential equations. In this talk an overview on the construction of exponential integrators will be given, implementation issues will be discussed, and examples of the methods applied to some well known problems like the nonlinear Schroedinger, the Kuramoto-Shivashinski and the Gray-Scott equations will be presented.