Introduction to the contents of issue 49:2

Abstract

We start this issue with a thorough contribution to a well studied problem in the numerical treatment of differential equations. In the first paper, Ben Adcock describes a spectral-Galerkin method for a two point boundary value problem with homogeneous Neumann boundary conditions. Modified Fourier series are used. In another differential equation paper, Etienne Emmrich solves a nonlinear evolution equation by a two step backward differentiation, BDF, formula, where the step is allowed to vary as long as the quotient between two subsequent steps is bounded. We are happy to include two contributions that deal with how to handle random effects when following a differential system forwards in time. Matthias Geissert, Mihaly Kova, and Stig Larsson solve a stochastic heat equation driven by additive noise, with a standard finite element discretization in space. It is shown to give a weak convergence at twice the rate of its strong convergence. In the next paper, Juan Carlos Jimenez and Felix Carbonell study the convergence of a local linearization method for a random differential system driven by commutative noise. A general theorem giving the order of convergence is given. It is applied to some implementations of local linearization algorithms. We have also two papers on initial value algorithms tailored to practically important applied problems. Ujjwal Koley, Siddharta Mishra, Nils Henrik Risebro, and Magnus Svard describe high order finite difference methods for the magnetic induction equations. The finite difference schemes are based on summation by parts operators for spatial derivatives and a simultaneous approximation term technique for imposing boundary conditions. Jorg Wensch, Oswald Knoth, and Alexander Galant treat the Euler equations of atmospheric flow. Such a flow contains both fast sound