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The aim of this paper is to analyze explicit exponential Runge-Kutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonstiff) order(More)
The Lanczos method is an iterative procedure to compute an orthogonal basis for the Krylov subspace generated by a symmetric matrix A and a starting vector v. An interesting application of this method is the computation of the matrix exponential exp(−τ A)v. This vector plays an important role in the solution of parabolic equations where A results from some(More)
In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigen-values with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical(More)
We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order(More)
We introduce a new class of exponential integrators for the numerical integration of large-scale systems of stiff differential equations. These so-called Rosenbrock-type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the Jacobian. In contrast to standard integrators, the methods are fully(More)
A well-known problem in computing some matrix functions iteratively is a lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Assume, the matrix exponential of a given matrix times a given vector has to be computed. We interpret the sought after vector as a value of a vector(More)
In this paper, we analyse a family of exponential integrators for second-order differential equations in which high-frequency oscillations in the solution are generated by a linear part. Conditions are given which guarantee that the integrators allow second-order error bounds independent of the product of the step size with the frequencies. Our convergence(More)