Exponential Taylor methods: Analysis and implementation

For the time integration of semilinear systems of differential equations, a class of multiderivative exponential integrators is considered. The methods are based on a Taylor series expansion of the semilinearity about the numerical solution, the required derivatives are computed by automatic differentiation. Inserting these derivatives into the variation-of-constants formula results in an exponential integrator which requires the action of the exponential of an augmented Jacobian only.The convergence properties of such exponential integrators are analyzed, and potential sources of numerical instabilities are identified. In particular, it is shown that local linearization gives rise to better stability for stiff problems. A number of numerical experiments illustrate the theoretical results.