Preface to BIT 53:4
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Preface to BIT 53:4 Axel Ruhe
Published online: 14 November 2013 © Springer Science+Business Media Dordrecht 2013
This year, we wind up BIT with a selection of papers from our regular stream of contributions. I note that now we have quite a few manuscripts in the Approximation area, both geometric design and numerical cubature, as well as some contributions to Numerical Linear Algebra. Only one paper deals with properties of time stepping methods for differential equations. But rest assured, more is coming up next year! These are the papers: Assyr Abdulle, Gilles Vilmart, and Konstantinos Zygalakis investigate the stability properties of integrators for Itô stochastic differential equations. They derive a singly diagonally implicit Runge Kutta method of weak second order, and compares its stability region to existing variants. Mean square and asymptotic stability is considered, as well as different variants of A stability. A.K.B. Chand and P. Viswanathan study shape preservation aspects of cubic Hermite fractal interpolation functions. They get a desirable shape by putting constraints on values of function and derivative at the knot points. Scaling factors in the subintervals are chosen, to force the function to stay inside a rectangle. Raffaele D’Ambrosio, Ernst Hairer, and Christophe J. Zbinden study long term integration of a conservative dynamical system. They prove that G-symplecticity of a general linear method implies conjugate-symplecticity of the underlying one step method. Catterina Dagnino, Paola Lamberti, and Sara Remogna construct new cubature rules for 3D integrals, based on spline quasi-interpolants, expressed as linear combinations of scaled and translated boxes and local linear functionals. Nodes and weights are given, as well as error bounds.
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A. Ruhe ( ) Numerical Analysis Group, School of Science (SCI), Royal Institute of Technology (KTH), 10044, Stockholm, Sweden e-mail: [email protected]
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A. Ruhe
Erik Martin-Dorel, Guillaume Melquiond, and Jean-Michel Muller study floating point computation in the practically interesting case where some parts of the computation are done in a higher precision than the original data. These computations are often a part of a much larger process, and then predictability is of importance, in order to get reliable bounds on the error imposed. Ralf Hiptmair, Carlos Jerez-Hanckes, and Christoph Schwab study the Maxwell cavity source problem in the frequency domain. They use discretization by means of sparse tensor edge elements, in the case when a full tensor product gives rise to a basis of too high a dimension. Tsung-Ming Huang, Zhongxiao Jia, and Wen-Wei Lin project a quadratic eigenvalue problem on a subspace, giving Ritz approximations to eigenvalues and eigenvectors. When the angle between an eigenvector and the subspace gets small, the Ritz values converge to the eigenvalue. They introduce the use of refined Ritz vectors which are proved to converge towards an eigenvector. Zhongxiao Jia and Qian Zhang do a systematic study on how to choose drop tolerance to
