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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 eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.

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Abstract. Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equations. As a side result, we obtain error bounds for Galerkin-type Krylov methods for linear equations, namely the biconjugate gradient method and the full orthogonalization method. For Krylov approximations to matrix exponentials, we show superlinear error decay from relatively small iteration numbers onwards, depending on the geometry of the numerical range, the spectrum or the pseudospectrum. The convergence to exp( A)v is faster than that of corresponding Krylov methods for the solution of linear equations (I A)x = v, which usually arise in the numerical solution of sti ordinary di erential equations. We therefore propose a new class of time integration methods for large systems of nonlinear di erential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step.

Abstract. We study the numerical integration of large sti systems of di erential equations by methods that use matrix-vector products with the exponential or a related function of the Jacobian. For large problems, these can be approximated by Krylov subspace methods, which typically converge faster than those for the solution of the linear systems arising in standard sti integrators. The exponential methods also o er favorable properties in the integration of di erential equations whose Jacobian has large imaginary eigenvalues. We derive methods up to order 4 which are exact for linear constant-coe cient equations. The implementation of the methods is discussed. Numerical experiments with reaction-di usion problems and a time-dependent Schr odinger equation are included.

Abstract. 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 conditions for exponential RungeKutta methods, but the main interest of our paper lies in the stiff case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we finally construct methods that do not suffer from order reduction.

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Abstract. 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 form of discretization of an elliptic operator. In the present paper we will argue that for these applications the convergence behavior of this method can be unsatisfactory. We will propose a modified method that resolves this by a simple preconditioned transformation at the cost of an innerouter iteration. A priori error bounds are presented that are independent of the norm of A. This shows that the worst case convergence speed is independent of the mesh width in the spatial discretization of the elliptic operator. We discuss, furthermore, a posteriori error estimation and the tuning of the coupling between the inner and outer iteration. We conclude with several numerical experiments with the proposed method.