A stable high-order Spectral Difference method for hyperbolic conservation laws on triangular elements

Numerical schemes using piecewise polynomial approximation are very popular for high order discretization of conservation laws. While the most widely used numerical scheme under this paradigm appears to be the Discontinuous Galerkin method, the Spectral Difference scheme has often been found attractive as well, because of its simplicity of formulation and implementation. However, recently it has been shown that the scheme is not linearly stable on triangles. In this paper we present an alternate formulation of the scheme, featuring a new flux interpolation technique using Raviart–Thomas spaces, which proves stable under a similar linear analysis in which the standard scheme failed. We demonstrate viability of the concept by showing linear stability both in the semi-discrete sense and for time stepping schemes of the SSP Runge–Kutta type. Furthermore, we present convergence studies, as well as case studies in compressible flow simulation using the Euler equations.     

Comparison of multigrid algorithms for high‐order continuous finite element discretizations

We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h‐multigrid and p‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse

Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.     

On distributionally robust multiperiod stochastic optimization

This paper considers model uncertainty for multistage stochastic programs. The data and information structure of the baseline model is a tree, on which the decision problem is defined. We consider “ambiguity neighborhoods” around this tree as alternative models which are close to the baseline model. Closeness is defined in terms of a distance for probability trees, called the nested distance. This distance is appropriate for scenario models of multistage stochastic optimization problems as was demonstrated in Pflug and Pichler (SIAM J Optim 22:1–23, 2012). The ambiguity model is formulated as a minimax problem, where the the optimal decision is to be found, which minimizes the maximal objective function within the ambiguity set. We give a setup for studying saddle point properties of the minimax problem. Moreover, we present solution algorithms for finding the minimax decisions at least asymptotically. As an example, we consider a multiperiod stochastic production/inventory control problem with weekly ordering. The stochastic scenario process is given by the random demands for two products. We determine the minimax solution and identify the worst trees within the ambiguity set. It turns out that the probability weights of the worst case trees are concentrated on few very bad scenarios.     

Elliptic optimal control problems with <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-control cost and applications for the placement of control devices

Elliptic optimal control problems with L ¹-control cost are analyzed. Due to the nonsmooth objective functional the optimal controls are identically zero on large parts of the control domain. For applications, in which one cannot put control devices (or actuators) all over the control domain, this provides information about where it is most efficient to put them. We analyze structural properties of L ¹-control cost solutions. For solving the non-differentiable optimal control problem we propose a semismooth Newton method that can be stated and analyzed in function space and converges locally with a superlinear rate. Numerical tests on model problems show the usefulness of the approach for the location of control devices and the efficiency of our algorithm.     

Studies of the protein and the energy metabolism in man during a wintering in Antarctica

During the 29th Soviet Antarctic Expedition in Novolazarevskaya from March 1984 to March 1985, the protein and energy metabolisms were studied in six expeditioners from the German Democratic Republic. The investigations were carried out at the beginning of the expedition (May), during the polar night (July) and during the polar day (December). The effect of a special stress situation (sledge trek in April 1984) was investigated in one subject. The stable nitrogen isotope (15)N was used to study the protein metabolism. The assessment of the energy metabolism was based on the oxygen consumption, which was determined by means of a spirograph. In addition, the vital capacity, the breath minute volume, the blood pressure, etc. were measured. The following results were obtained: During the polar night, the utilisation of the dietary proteins and the whole body protein synthesis calculated by means of the (15)N excretion of the total nitrogen in urine were greater (73.6±0.9 % and 3.48±0.17?g protein d(-1)?kg(-1), n=3) than the respective values during the polar day (69.7±1.2, p<0.05, n=3 and 3.05±0.07, p<0.05, n=3) and at the beginning of the expedition (69.6±1.4, p<0.02, n=5 and 2.81±0.09, p<0.01, n=5). The lowest values (58.0 % and 2.43?g protein d(-1)?kg(-1)) were obtained in the subject after the trek. The resting metabolic rate (in kJ?d(-1)?m(-2)) was decreased during the polar night (45.6±5.0, n=4) in comparison with the polar day (61.5±11.3, n=3) and the beginning of the expedition (52.3±9.6, n=4) with p<0.01 in both cases.     

Parametrically shaped femtosecond pulses in the nonlinear regime obtained by reverse propagation in an optical fiber

We present the experimental realization of a method to generate predetermined, arbitrary pulse shapes after transmission through an optical fiber in the nonlinear regime. The method is based on simulating the reverse propagation of the desired pulse shape in the fiber. First, linear and nonlinear parameters of a single-mode step-index fiber required for the simulation are determined. The calculated pulse shapes are then generated in a pulse shaper.     

Thermal decomposition of trimethylgallium Ga(CH3)3: a shock-tube study and first-principles calculations

The thermal decomposition of Ga(CH3)3 has been studied both experimentally in shock-heated gases and theoretically within an ab-initio framework. Experiments for pressures ranging from 0.3 to 4 bar were performed in a shock tube equipped with atomic resonance absorption spectroscopy (ARAS) for Ga atoms at 403.3 nm. Time-resolved measurements of Ga atom concentrations were conducted behind incident waves as well as behind reflected shock waves at temperatures between 1210 and 1630 K. The temporal variation in Ga-atom concentration was described by a reaction mechanism involving the successive abstraction of methyl radicals from Ga(CH3)3 (R1), Ga(CH3)2 (R2), and GaCH3 (R3), respectively, where the last reaction is the rate-limiting step leading to Ga-atom formation. The rate constant of this reaction (R3) was deduced from a simulation of the measured Ga-atom concentration profiles using thermochemical data from ab-initio calculations for the reactions R1 and R2 as input. The Rice-Ramsperger-Kassel-Marcus (RRKM) method including variational transition state theory was applied for reaction R3 assuming a loose transition state. Structural parameters and vibrational frequencies of the reactant and transition state required for the RRKM calculations were obtained from first-principles simulations. The energy barrier E3(0) of reaction R3, which is the most sensitive parameter in the calculation, was adjusted until the RRKM rate constant matched the experimental one and was found to be E(0) = 288 kJ/mol. This value is in a good agreement with the corresponding ab-initio value of 266 kJ/mol. The rate constant of reaction R3 was found to be k 3/(cm(3) mol(-1)s(-1)) = 2.34 x 10(11) exp[-23330(K/ T)].