A short course on exponential integrators

This paper contains a short course on the construction, analysis , and implementation of exponential integrators for time dependent partial differential equations. A much more detailed recent review can be found in Hochbruck and Ostermann (2010). Here, we restrict ourselves to one-step methods for autonomous problems. A basic principle for the construction of exponential integra-tors is the linearization of a semilinear or a nonlinear evolution equation. We distinguish exponential Runge–Kutta methods, using a fixed linearization and exponential Rosenbrock-type methods , which use a continuous linearization at the current approximation of the solution. We present some of the convergence results and give a proof for the simplest method, the exponential Euler method. The fact that it is possible to construct explicit exponential integrators which obey error bounds even for abstract evolution equations comes at the price that one has to approximate products of matrix functions with vectors in the spatially discrete case. For an efficient implementation one has to combine the integrator with well-chosen algorithms from numerical linear algebra. We briefly sketch Krylov subspace methods for this task.