The paper is concerned with the construction, implementation and numerical analysis of exponential multistep methods. These methods are related to explicit Adams methods but, in contrast to the latter, make direct use of the exponential and related matrix functions of a (possibly rough) linearization of the vector field. This feature enables them to integrate stiff problems explicitly in time.
A stiff error analysis is performed in an abstract framework of linear semigroups that includes semilinear evolution equations and their spatial discretizations. A possible implementation of the proposed methods, including the computation of starting values and the evaluation of the arising matrix functions by Krylov subspace methods is discussed. Moreover, an interesting connection between exponential Adams methods and a class of local time stepping schemes is established.
Numerical examples that illustrate the methods' properties are included.