a b s t r a c tWe give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrödinger, (linear) time-dependent Schrödinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.

Mike A. Botchev devotes this work to the memory of his father. Abstract.A well-known problem in computing some matrix functions iteratively is the lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Suppose the matrix exponential of a given matrix times a given vector has to be computed. We develop the approach of Druskin, Greenbaum, and Knizhnerman [SIAM J. Sci. Comput., 19 (1998), pp. 38-54] and interpret the sought-after vector as the value of a vector function satisfying the linear system of ordinary differential equations (ODEs) whose coefficients form the given matrix. The residual is then defined with respect to the initial value problem for this ODE system. The residual introduced in this way can be seen as a backward error. We show how the residual can be computed efficiently within several iterative methods for the matrix exponential. This resolves the question of reliable stopping criteria for these methods. Further, we show that the residual concept can be used to construct new residual-based iterative methods. In particular, a variant of the Richardson method for the new residual appears to provide an efficient way to restart Krylov subspace methods for evaluating the matrix exponential.

Abstract. In this paper we analyse a family of exponential integrators for secondorder 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 analysis generalises known results on the mollified impulse method by García-Archilla, Sanz-Serna and Skeel [6] and on Gautschi-type exponential integrators [12,13].

Abstract.We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

This paper studies a numerical method for second-order oscillatory differential equations in which high-frequency oscillations are generated by a linear time-and/or solution-dependent part. For constant linear part, it is known that the method allows second-order error bounds independent of the product of the step-size with the frequencies and is therefore a long-time-step method. Most real-world problems are not of that kind and it is important to study more general equations. The analysis in this paper shows that one obtains second-order error bounds even in the case of a time-and/or solution-dependent linear part if the matrix is evaluated at averaged positions. (2000): 65L05, 65L70 Mathematics Subject Classification