A generic framework for time-stepping partial differential equations (PDEs): general linear methods,object-oriented implementation and application to fluid problems

Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework – both in terms of algorithms and implementations – that allows an almost seamless switch between various explicit, implicit and implicit–explicit (IMEX) time-stepping methods. We put particular emphasis on how to incorporate time-dependent boundary conditions, an issue that goes beyond classical ODE theory but which plays an important role in the time-stepping of the PDEs arising in computational fluid dynamics. Our algorithm is based upon J.C. Butcher's unifying concept of general linear methods that we have extended to accommodate the family of IMEX schemes that are often used in engineering practice. In the article, we discuss design considerations and present an object-oriented implementation. Finally, we illustrate the use of the framework by applications to a model problem as well as to more complex fluid problems.     

An α-Lactam as an Alkilating Agent

The ‘normal’ reaction of α-lactams (aziridinones) is mcleophilic ring ening,¹ which can occur with fission of either one of the three bonds (a-b-c), depending on the reagent, to yield and acyclic addition prodnct (I,II,III).²     

Recent advances in the homogeneous polymerisation of olefins mediated by nickel complexes

A brief overview is presented on recent advances in the application of nickel coordination complexes as mediators of olefin polymerisation; in some cases involving stereo-regular polymer formation where appropriate. Emphasis is also given on recent findings concerning di- and tri-nuclear Ni clusters with details on ethylene, methyl methacrylate or styrene-based monomer polymerisations.     

Modal and characteristics-based approaches for modeling elastic waves induced by time-dependent boundary conditions

In this paper, we present a characteristics-based approach for solving elastic wave problems with time-dependent traction boundary conditions. A generalized mathematical model for this important class of problems is expressed as a set of first-order, linear, hyperbolic partial differential equations. We analyze the mathematical structure of this first-order linear system, verify its hyperbolicity, derive its characteristic form, and deduce its eigenvalues, eigenvectors, and Riemann invariants. The eigenvalues correspond to the wave speeds, while the Riemann invariants are used to construct a solution by the method of characteristics.     

Fundamental problems of solid mechanics in high technologies

The paper provides an analytical review of some fundamental problems of solid mechanics in high technologies. The review relies partly on the invited report presented by the authors at the 10th All-Russian Congress on Theoretical and Applied Mechanics held in Nizhny Novgorod in 2011. Attention is focused on the problems involved in technologies of materials with a bulk fine-grained (nano) structure and in technologies used to control mechanical properties of surfaces and interfaces.