Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented.

Asymptotic analysis of combined breather-kink modes in a Fermi-Pasta-Ulam chain

We find approximations to travelling breather solutions of the one-dimensional Fermi-Pasta-Ulam (FPU) lattice. Both bright breather and dark breather solutions are found. We find that the existence of localised (bright) solutions depends upon the coefficients of cubic and quartic terms of the potential energy, generalising an earlier inequality derived by James [CR Acad Sci Paris 332, 581, (2001)]. We use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear Schr{\"o}dinger equation at leading order and hence construct an asymptotic form for the breather. We show that in the absence of a cubic potential energy term, the lattice supports combined breathing-kink waveforms. The amplitude of breathing-kinks can be arbitrarily small, as opposed to traditional monotone kinks, which have a nonzero minimum amplitude in such systems. We also present numerical simulations of the lattice, verifying the shape and velocity of the travelling waveforms, and confirming the long-lived nature of all such modes.

Fundamental behaviour of Virtual-Build-to-Order systems

Virtual-Build-to-Order (VBTO) is an emerging order fulfilment system within the automotive sector that is intended to improve fulfilment performance by taking advantage of integrated information systems. The primary innovation in VBTO systems is the ability to make available all unsold products that are in the production pipeline to all customers. In a conventional system the pipeline is inaccessible and a customer can be fulfilled by a product from stock or having a product Built-to-Order (BTO), whereas in a VBTO system a customer can be fulfilled by a product from stock, by being allocated a product in the pipeline, or by a build-to-order product. Simulation is used to investigate and profile the fundamental behaviour of the basic VBTO system and to compare it to a Conventional system. A predictive relationship is identified, between the proportions of customers fulfilled through each mechanism and the ratio of product variety / pipeline length. The simulations reveal that a VBTO system exhibits inherent behaviour that alters the stock mix and levels, leading to stock levels being higher than in an equivalent conventional system at certain variety / pipeline ratios. The results have implications for the design and management of order fulfilment systems in sectors such as automotive where VBTO is a viable operational model.