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In recent years, a great deal of attention has been focused on numericallysolving exponential integrators. The important ingredient to the implementationof exponential integrators is the efficient and accurate evaluation of the socalled $\varphi$-functions on a given vector. The Krylov subspace method is animportant technique for this problem. For this type of method, however,restarts become essential for the sake of storage requirements or due to thegrowing computational complexity of evaluating the matrix function on aHessenberg matrix of growing size. Another problem in computing$\varphi$-functions is the lack of a clear residual notion. The contribution ofthis work is threefold. First, we introduce a framework of the harmonic Arnoldimethod for $\varphi$-functions, which is based on the residual and the obliqueprojection technique. Second, we establish the relationship between theharmonic Arnoldi approximation and the classical Arnoldi approximation, andcompare the harmonic Arnoldi method with the Arnoldi method from a theoreticalpoint of view. Third, we apply the thick-restarting strategy to the harmonicArnoldi method, and propose a thick-restated harmonic Arnoldi algorithm forevaluating $\varphi$-functions. An advantage of the new algorithm is that wecan compute several $\varphi$-functions simultaneously in the same searchsubspace. We show the merit of augmenting approximate eigenvectors in thesearch subspace, and give insight into the relationship between the error andthe residual of $\varphi$-functions. Numerical experiments show the superiorityof our new algorithm over many state-of-the-art algorithms for the computationof $\varphi$-functions.

A framework of the harmonic Arnoldi method for evaluating $\varphi$-functions with applications to exponential integrators