Preface to BIT 53:3

Abstract

In this penultimate number for 2013, we return to the areas most often covered by BIT, with one linear algebra paper, two contributions to approximation and the remaining eight dealing with differential equations, algorithms for time stepping, finite elements and stochastic equations. I expect few of my readers to go through it all, as I do, but it is interesting to see trends in the BIT subplot of the scientific field. These are the papers: John Carroll and Eoin O’Callaghan describe an exponential almost Runge-Kutta method, which is exact for a linear system of ordinary differential equations. It involves following values also of derivative approximations, but the advantage is that it is stable as an implicit method. Its behavior is demonstrated on time stepping for two parabolic problems, one Brusselator and one reaction diffusion advection equation. Stefan Guttel and Leonid Knizhnerman describe a rational Arnoldi method to compute a matrix function applied to a vector. It is applicable to the class of CauchyStieltjes functions, which contains square roots, logarithms and solutions to certain linear differential equations. It contains an automatic selection of the poles of the rational function and a residual based error estimator. Its behavior is demonstrated on some highly nonnormal test cases, as well as on an applied problem, computing the impedance function of a convection diffusion problem. Yoshio Komori and Evelyn Buckwar describe how to embed a deterministic RungeKutta method with high order, into a weak order stochastic Runge-Kutta method, and use this to solve a non-commutative stochastic differential equation. Derivative free methods are of special interest, and the method is compared to other available stochastic Runge-Kutta schemes.