A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge--Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive algorithm

Orientational order parameters of a de Vries–type ferroelectric liquid crystal obtained by polarized Raman spectroscopy and x-ray diffraction

The orientational order parameters 〈P2〉 and 〈P4〉 of the ferroelectric, de Vries–type liquid crystal 9HL have been determined in the SmA* and SmC* phases by means of polarized Raman spectroscopy, and in the SmA* phase using x-ray diffraction. Quantum density functional theory predicts Raman spectra for 9HL that are in good agreement with the observations and indicates that the strong Raman band probed in the experiment corresponds to the uniaxial, coupled vibration of the three phenyl rings along the molecular long axis. The magnitudes of the orientational order parameters obtained in the Raman and x-ray experiments differ dramatically from each other, a discrepancy that is resolved by considering that the two techniques probe the orientational distributions of different molecular axes. We have developed a systematic procedure in which we calculate the angle between these axes and rescale the orientational order parameters obtained from x-ray scattering with results that are then in good agreement with the Raman data. At least in the case of 9HL, the results obtained by both techniques support a “sugar loaf” orientational distribution in the SmA* phase with no qualitative difference to conventional smectics A. The role of individual molecular fragments in promoting de Vries–type behavior is considered

Field-Dependent Tilt and Birefringence of Electroclinic Liquid Crystals: Theory and Experiment

An unresolved issue in the theory of liquid crystals is the molecular basis of the electroclinic effect in the smectic-A phase. Recent x-ray scattering experiments suggest that, in a class of siloxane-containing liquid crystals, an electric field changes a state of disordered molecular tilt in random directions into a state of ordered tilt in one direction. To investigate this issue, we measure the optical tilt and birefringence of these liquid crystals as functions of field and temperature, and we develop a theory for the distribution of molecular orientations under a field. Comparison of theory and experiment confirms that these materials have a disordered distribution of molecular tilt directions that is aligned by an electric field, giving a large electroclinic effect. It also shows that the net dipole moment of a correlated volume of molecules, a key parameter in the theory, scales as a power law near the smectic-A--smectic-C transition.Comment: 18 pages, including 9 postscript figures, uses REVTeX 3.0 and epsf.st

Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data

We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control

Ferroelectric ordering in chiral smectic C^* liquid crystals determined by nonchiral intermolecular interactions

General microscopic mechanism of ferroelectric ordering in chiral smectic C* liquid crystals is considered. It is shown that if the mesogenic molecules have a sufficiently low symmetry, the spontaneous polarization is proportional to one of the biaxial vector order parameters of the smectic C phase. This order parameter may be determined by intermolecular interactions which are not sensitive to molecular chirality. At the same time, the polarization is also proportional to a pseudoscalar parameter which vanishes if the molecules are nonchiral. The general statistical theory of ferroelectric ordering is illustrated by two particular models. The first model is based on electrostatic quadrupole-quadrupole interactions, and it enables one to obtain explicit analytical expressions for the spontaneous polarization. In the second model, the molecular chirality and polarity are determined by a pair of off-center nonparallel dipoles. For this case, the spontaneous polarization is calculated numerically as a function of temperature. The theory provides a more general interpretation of the previous approaches including the classical Boulder model

The peculiar optic, dielectric and x-ray diffraction properties of a fluorinated de vries asymmetric-diffuse-cone-model ferroelectric liquid crystal

A new semi-fluorinated chiral smectic liquid crystal, W504, is investigated by electro-optic, dielectric and X-ray scattering experiments. It exhibits a huge dielectric soft mode response, strong electroclinic effect and a birefringence which increases considerably with the director tilt angle theta; typical characteristics of a SmA - SmC transition following the de Vries asymmetric diffuse cone (ADC) model in which the non-zero director tilt in SmC arises through an ordering of tilting directions rather than an actual increase in average molecule tilt . In W504 a small increase in of about 4 degrees is however detected in the SmC phase. Although the increase in molecule inclination is much less than the increase in director tilt h, saturating close to 30 degrees, it leads to a shrinkage of the smectic layers by about 1 angstrom, a result of the large initial molecule tilt in the SmA phase, (SmA) approximate to 30 degrees. The tilting transition in W504 is thus mainly an ADC model disorder - order transition, but it also has a component of a structural transition. The semi- fluorinated molecular structure of W504 leads to a very weak electron density modulation along the layer normal, giving a vanishing form factor in bulk samples which exhibit no (001) X-ray scattering peak. In thin films the (001) peak is however observed, indicating that the electron density modulation is enhanced by the breaking of the head - tail symmetry of the liquid crystal phase at the LC - air interface