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.

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Modelling, Analysis and Simulation Modelling, Analysis and SimulationNumerical integration of damped Maxwell equations M.A. Botchev, J.G. Verwer Numerical integration of damped Maxwell equations ABSTRACT We study the numerical time integration of Maxwell's equations from electromagnetism. Following the method of lines approach we start from a general semi-discrete Maxwell system for which a number of time-integration methods are considered. These methods have in common an explicit treatment of the curl terms. Central in our investigation is the question how to efficiently raise the temporal convergence order beyond the standard order of two, in particular in the presence of an explicitly or implicitly treated damping term which models conduction. REPORT MAS-E0804 APRIL 2008 AbstractWe study the numerical time integration of Maxwell's equations from electromagnetism. Following the method of lines approach we start from a general semi-discrete Maxwell system for which a number of time-integration methods are considered. These methods have in common an explicit treatment of the curl terms. Central in our investigation is the question how to efficiently raise the temporal convergence order beyond the standard order of two, in particular in the presence of an explicitly or implicitly treated damping term which models conduction.

Computational codes for direct numerical simulations of Rayleigh-Bénard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra = 10 8 and Pr = 1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBflow) and a specialized fourth-order finite-volume code (Goldfish) are compared with a general purpose finite-volume approach (OpenFOAM) and a general purpose spectral-element code (Nek5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBflow and Goldfish are found to excel in efficiency, outperforming the general purpose flow solvers Nek5000 and OpenFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using Nek5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.