Exponential-Krylov methods for ordinary differential equations

This paper develops a new family of exponential time discretization methods called exponential-Krylov (expK). The new schemes treat the time discretization and the Krylov-based approximation of exponential matrix-vector products as a single computational process. The classical order conditions theory developed herein accounts for both the temporal and the Krylov approximation errors. Unlike traditional exponential schemes, expK methods require the construction of only a single Krylov space at each timestep. The number of basis vectors that guarantee the temporal order of accuracy does not depend on the application at hand. Numerical results show favorable properties of expK methods when compared to current exponential schemes.