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A detailed description of GENE-3D, a newly developed global stellarator version of the well established gyrokinetic turbulence code GENE, is provided – along with some initial simulation results. First, the underlying gyrokinetic equations and the use of field-aligned coordinates in non-axisymmetric magnetohydrodynamic equilibria are discussed. Then, various aspects regarding the numerical implementation in GENE-3D are described. Finally, the code is validated and its parallel performance is assessed, along with the influence of numerical precision.

The impact of dynamic models for applications to LES of compressible flows is assessed in the framework of a numerical model based on high order discontinuous finite elements. The projections onto lower dimensional subspaces associated with lower degree basis functions are used as LES filter, along the lines proposed in Variational Multiscale templates. Comparisons with DNS results available in the literature for plane and constricted channel flows at Mach numbers 0.2, 0.7 and 1.5 show clearly that the dynamic models are able to improve the prediction of most key features of the flow with respect to the Smagorinsky models employed so far in a VMS-DG context.

We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov–Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [4] , with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss–Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge–Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau–Lifshitz–Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes since the only polynomial function satisfying the unit sphere constraint everywhere are constants. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. A priori energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both a priori energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. This is due to the fact that the unit length at the nodes is not satisfied in general when a time integration is performed. We will carry out a linear Euler time-stepping method and a non-linear Crank–Nicolson method. The latter is solved by using the former as a non-linear solver. Ministerio de Economía y Competitividad MTM2015-69875-P

We perform in this paper the numerical analysis of some penalty stabilized solvers for the unsteady Navier---Stokes equations. We consider low-order and high-order methods. The low-order method is a pure penalty method, while the high-order one is a projection-stabilized method. We perform their numerical analysis (stability and convergence) for solutions that only need to bear the natural regularity. In this analysis, the stability is based upon specific inf-sup conditions. No local orthogonality properties are needed for the projection-interpolation operator. The convergence is based upon the representation of the stabilizing terms by means of bubble finite element spaces. We include some numerical tests for realistic flows that confirm the theoretical expectations.

A dynamic model for a fixed-bed reactor for methanol synthesis is presented. The model is compared with its steady state version. The analysis points out that the numerical stability of the dynamic model is improved by opportunely increasing the level of detail. It is appropriate to introduce the diffusion terms, to work with mass fractions, to select good discretization methods for each term of the model equations. Since these aspects are usually neglected in steady state analysis, this paper investigates step-by-step their implementation, emphasizing their importance (I) in the transformation of an original hyperbolic PDE system into a parabolic PDE system; (II) in removing non-physical oscillations generated by first-order systems that may lead to relevant model prediction errors; and (III) in the approximation of the convection terms using the forward formulation, which is more stable and provides more realistic solutions.

We consider the finite element discretization of a convection-diffusion equation, where the convection term is handled via a fluctuation splitting algorithm. We prove a posteriori error estimates which allow us to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach. Nous considérons une discrétisation par éléments finis d’une équation de convection-diffusion, où un algorithme de décentrage est utilisé pour traiter le terme de convection. Nous prouvons des estimations d’erreur a posteriori qui permettent d’adapter le maillage pour optimiser la discrétisation de ces équations. Des résultats numériques confirment l’intérêt d’une telle approche. Junta de Andalucía

This paper proposes and compares different steady-state models for the methanol synthesis fixed-bed reactor. The main goal is to indicate the pros and cons of these alternative models, to highlight their peculiarities in providing accurate and reliable estimations, and to warn the reader about possible risks involved in certain modeling assumptions. The Lurgi-type shell and tube boiler-reactor mathematical modeling is also laid out, before more complex issues and research, such as dynamic simulation or PDE-based predictive control, are broached.

Non-hydrostatic Unified Model of the Atmosphere (NUMA) The first NUMA papers appeared in 2008. From 2008 through 2010, all the NUMA papers appearing involved the 2D (x-z slice) Euler equations. All the theory and numerical implementations were first developed in 2D. The article of record may be found at http://dx.doi.org/10.1137/070708470 A discontinuous Galerkin (DG) finite element formulation is proposed for the solu- tion of the compressible Navier–Stokes equations for a vertically stratified fluid, which are of interest in mesoscale nonhydrostatic atmospheric modeling. The resulting scheme naturally ensures con- servation of mass, momentum, and energy. A semi-implicit time-integration approach is adopted to improve the efficiency of the scheme with respect to the explicit Runge–Kutta time integration strategies usually employed in the context of DG formulations. A method is also presented to refor- mulate the resulting linear system as a pseudo-Helmholtz problem. In doing this, we obtain a DG discretization closely related to those proposed for the solution of elliptic problems, and we show how to take advantage of the numerical integration rules (required in all DG methods for the area and flux integrals) to increase the efficiency of the solution algorithm. The resulting numerical for- mulation is then validated on a collection of classical two-dimensional test cases, including density driven flows and mountain wave simulations. The performance analysis shows that the semi-implicit method is, indeed, superior to explicit methods and that the pseudo-Helmholtz formulation yields further efficiency improvements.

In this article, we propose a novel discontinuous Galerkin method for convection-diffusion-reaction problems, characterized by three main properties. The first is that the method is hybridizable; t...

Non-hydrostatic Unified Model of the Atmosphere (NUMA) The first NUMA papers appeared in 2008. From 2008 through 2010, all the NUMA papers appearing involved the 2D (x-z slice) Euler equations. All the theory and numerical implementations were first developed in 2D. We present spectral element (SE) and discontinuous Galerkin (DG) solutions of the Euler and compressible Navier– Stokes (NS) equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric modeling. We study three different forms of the governing equations using seven test cases. Three test cases involve flow over moun- tains which require the implementation of non-reflecting boundary conditions, while one test requires viscous terms (den- sity current). Including viscous stresses into finite difference, finite element, or spectral element models poses no additional challenges; however, including these terms to either finite volume or discontinuous Galerkin models requires the introduc- tion of additional machinery because these methods were originally designed for first-order operators. We use the local discontinuous Galerkin method to overcome this obstacle. The seven test cases show that all of our models yield good results. The main conclusion is that equation set 1 (non-conservation form) does not perform as well as sets 2 and 3 (con- servation forms). For the density current (viscous), the SE and DG models using set 3 (mass and total energy) give less dissipative results than the other equation sets; based on these results we recommend set 3 for the development of future multiscale research codes. In addition, the fact that set 3 conserves both mass and energy up to machine precision motives us to pursue this equation set for the development of future mesoscale models. For the bubble and mountain tests, the DG models performed better. Based on these results and due to its conservation properties we recommend the DG method. In the worst case scenario, the DG models are 50% slower than the non-conservative SE models. In the best case scenario, the DG models are just as efficient as the conservative SE models.

In this article, we devise a simulation tool of the dynamics of a tethered buoy, which is a mooring system consisting of a rigid floating body (buoy) connected by an elastic cable (mooring line) to the bottom of the fluid environment. The accuracy and robustness of the overall solution algorithm are based on an appropriate choice of three main ingredients. The first ingredient is the use of a novel mixed finite element formulation for the mooring line, which allows a robust modeling of the cable even in the limit of an infinite value of the Young modulus. The second ingredient is the use of quaternions in the equations of motion of the floating body, instead of Euler angles, which avoids angle singularities in the presence of large rotations. The third ingredient is the use of an implicit method (Backward Euler) for time discretization, in conjunction with a damped Newton method for the solution of the resulting nonlinear system, which avoids computational inefficiency of explicit time integration schemes in the presence of quasi-inextensible cables and a fine spatial grid size. The physical soundness of the mooring system model and the accuracy and robustness of the proposed numerical procedure are validated in the study of a tethered buoy operating under static and dynamic industrial-like working conditions.

A two dimensional mesoscale atmospheric model is presented based on two systems of dynamical equations, the non-hydrostatic compressible Euler equations and the corresponding hydrostatic system. Both equation systems are discretized with a discontinuous Galerkin method in space and a linear semi-implicit multistep method in time. To cover elementwise polynomial spaces up to order four, exact quadrature rules are used. The method is applied to a terrain following quadrilateral grid. To validate the models, gravity wave propagation and mountain wave experiments are analyzed. The numerical models exhibit the proper wave propagation characteristics and high order convergence rates.

We present semi-implicit (implicit-explicit) formulations of the compressible Navier-Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit Runge-Kutta method for all the tests studied, especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including inertia-gravity waves, density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of nonreflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and nonreflecting boundary conditions (NRBCs). It is shown that the NRBCs exert a strong impact on the accuracy and efficiency of the models. Office of Naval Research Junta de Andalucía German Research Foundation

In this work we deal with the numerical simulation of thermal oxidation in silicon device technology. This application involves the coupled solution of a diffusion–reaction problem and of a mechanical problem. These two problems are mutually dependent through the exchange of stresses and fluxes that are typically post-processed fields in standard finite element approaches, and as such they may suffer from a lack of accuracy and from physical inconsistencies. In this paper, we propose a novel numerical approach to the simulation of the thermal oxidation process, that is characterized by the use of mixed and hybrid finite elements. The main advantage of such formulations is that stresses and fluxes are directly computed quantities, rather than obtained from post-processing techniques. We also address the procedures and the techniques that must be devised for handling the coupled interaction problem and the presence of a computational grid moving in time. The numerical approach we propose is eventually validated on a realistic example of the thermal oxidation process in a local oxidation structure (LOCOS).