Exponential Integrators for Semilinear Problems

In the present work, exponential integrators for time integration of semilinear problems are studied. These integrators, as there name suggests, use the exponential and often functions which are closely related to the exponential function inside the numerical method. Three main classes of exponential integrators, exponential linear multistep (multivalue), exponential Runge–Kutta (multistage) and exponential general linear methods, are discussed. A general formulation of exponential integrators, which includes, as special cases, all known methods, is proposed. The nonstiff order theory for exponential multistage methods, along with a non-recursive rule for generating each order condition from its corresponding rooted tree, is derived. The natural connection between exponential integrators and Lie group methods with affine algebra action is also studied. A new approach for deriving Generalized Integrating Factor Runge–Kutta methods, which allows the nonlinear part of the problem to be approximated by trigonometric polynomials, is proposed. The crucial role of the algebra action in the overall performance of any Lie group method is discussed. A new algebra action arising from the solutions of differential equations with nonautonomous frozen vector fields is proposed. The corresponding exponential integrators based on this action are derived. Different methods for numerically stable computation of the most commonly used functions which appear in the format of an exponential integrator are considered. A generalization of the method based on the tridiagonal reduction is proposed. The new approach allows to compute all functions included in the format of an exponential integrator in the case when the arguments are symmetric (Hermitian) matrices. Some practical issues regarding variable step size implementations as well as the main advantages and disadvantages of the considered numerical techniques are discussed. New effective methods and their modifications for solving special three and five diagonal block systems of linear equations, based on a modified LU factorization are proposed. For illustrating the theoretical results, several numerical experiments are presented.