CFL Conditions for Runge–Kutta discontinuous Galerkin methods on triangular grids

We study time step restrictions due to linear stability constraints of Runge–Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax–Friedrichs flux or the upwind flux, and integrated in time with various Runge–Kutta schemes designed for linear wave propagation problems or non-linear applications. Von–Neumann-like analyses are performed on structured periodic grids made up of congruent elements, to investigate the influence of element shape on the stability restrictions. We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height prove appropriate, although they are not totally independent of the triangle shape. We explain their general behaviour with respect to element quality, and report the corresponding Courant numbers with both types of flux and polynomial order p ranging from 1 to 10, for use as guidelines in practical simulations. We also compare the performance of the Lax–Friedrichs flux and the upwind flux, and we draw general conclusions about the relative computational efficiency of RK schemes. The application of CFL conditions to two examples involving respectively an unstructured and a hybrid grid confirms our results, although it shows that local stability criteria tend to yield too restrictive conditions.     

A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation

To develop and analyze new computational techniques for the Boltzmann equation based on model or approximation adaptivity, it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the Boltzmann equation and that admits the extraction of highly accurate reference solutions. For standard collision processes, the Boltzmann equation itself fails to meet the second requirement for d = 2, 3 spatial dimensions, on account of its setting in 2d, while for d = 1 the first requirement is violated because the Boltzmann equation then lacks the convergence-to-equilibrium property that forms the substructure of simplified models. In this article we present a numerical investigation of a new one-dimensional prototype of the Boltzmann equation. The underlying molecular model is endowed with random collisions, which have been fabricated such that the corresponding Boltzmann equation exhibits convergence to Maxwell–Boltzmann equilibrium solutions. The new Boltzmann model is approximated by means of a discontinuous Galerkin (DG) finite-element method. To validate the one-dimensional Boltzmann model, we conduct numerical experiments and compare the results with Monte-Carlo simulations of equivalent molecular-dynamics models.     

Physalis method for heterogeneous mixtures of dielectrics and conductors: Accurately simulating one million particles using a PC

Prosperetti’s seminal Physalis method, an Immersed Boundary/spectral method, had been used extensively to investigate fluid flows with suspended solid particles. Its underlying idea of creating a cage and using a spectral general analytical solution around a discontinuity in a surrounding field as a computational mechanism to enable the accommodation of physical and geometric discontinuities is a general concept, and can be applied to other problems of importance to physics, mechanics, and chemistry. In this paper we provide a foundation for the application of this approach to the determination of the distribution of electric charge in heterogeneous mixtures of dielectrics and conductors. The proposed Physalis method is remarkably accurate and efficient. In the method, a spectral analytical solution is used to tackle the discontinuity and thus the discontinuous boundary conditions at the interface of two media are satisfied exactly. Owing to the hybrid finite difference and spectral schemes, the method is spectrally accurate if the modes are not sufficiently resolved, while higher than second-order accurate if the modes are sufficiently resolved, for the solved potential field. Because of the features of the analytical solutions, the derivative quantities of importance, such as electric field, charge distribution, and force, have the same order of accuracy as the solved potential field during postprocessing. This is an important advantage of the Physalis method over other numerical methods involving interpolation, differentiation, and integration during postprocessing, which may significantly degrade the accuracy of the derivative quantities of importance. The analytical solutions enable the user to use relatively few mesh points to accurately represent the regions of discontinuity. In addition, the spectral convergence and a linear relationship between the cost of computer memory/computation and particle numbers results in a very efficient method. In the present paper, the accuracy of the method is numerically investigated by example computations using one dielectric particle, one isolated conductor particle, one conductor particle connected to an external source with imposed voltage, and two conductor/dielectric particles with strong interactions. The efficiency of the method is demonstrated with one million particles, which suggests that the method can be used for many important engineering applications of broad interest.     

Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

We study a family of generalized slope limiters in two dimensions for Runge–Kutta discontinuous Galerkin (RKDG) solutions of advection-diffusion systems. We analyze the numerical behavior of these limiters applied to a pair of model problems, comparing the error of the approximate solutions, and discuss each limiter’s advantages and disadvantages. We then introduce a series of coupled p-enrichment schemes that may be used as standalone dynamic p-enrichment strategies, or may be augmented via any in the family of variable-in-p slope limiters presented.     

Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system

Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community , , , , , ,  and . In this paper, we propose a new Strang split SL discontinuous Galerkin (DG) method for solving the Vlasov equation. Specifically, we apply the Strang splitting for the Vlasov equation [6], as a way to decouple the nonlinear Vlasov system into a sequence of 1-D advection equations, each of which has an advection velocity that only depends on coordinates that are transverse to the direction of propagation. To evolve the decoupled linear equations, we propose to couple the SL framework with the semi-discrete DG formulation. The proposed SL DG method is free of time step restriction compared with the Runge–Kutta DG method, which is known to suffer from numerical time step limitation with relatively small CFL numbers according to linear stability analysis. We apply the recently developed positivity preserving (PP) limiter [37], which is a low-cost black box procedure, to our scheme to ensure the positivity of the unknown probability density function without affecting the high order accuracy of the base SL DG scheme. We analyze the stability and accuracy properties of the SL DG scheme by establishing its connection with the direct and weak formulations of the characteristics/Lagrangian Galerkin method [23]. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and via classical plasma problems, such as Landau damping and the two stream instability.     

Directional coupler design based on coupled cavity waveguides in photonic crystals

We propose a directional coupler design based on coupled cavity waveguide in photonic crystals. The plane wave expansion is used to give the dispersion of the coupled cavity waveguides and two parallel such waveguides. The couple length is got from the dispersion curves. Based on the research of the dispersion, we present a directional coupler and the transmission property is given. This structure is potentially important for highly efficient directional coupler in integrate optical circuit.     

A comparative study of dispersion relation of EM waves in ternary one-dimensional plasma photonic crystals having two different structures

In this paper, a rigorous theoretical analysis has been made to study the dispersion relation of EM waves in periodic ternary one-dimensional photonic crystal having two different structures. In one case we have chosen glass-plasma and ZnS in one unit cell and in other case we have considered glass-plasma and M_gF₂ in one unit cell. Using Kronig-Penney model the dispersion relation for proposed structures has been obtained and numerical results are presented in the form of dispersion curves. The dependence of photonic band gap (PBG) characteristics on plasma frequency, plasma width and the width of dielectric media are discussed in the light of frequency gap and cutoffs of binary one-dimensional plasma photonic crystal. An attempt has been made to show how the PBG characteristic of a particular structure changes when the dielectric materials of its unit cell is changed by the other dielectric material. It is found that the structure having glass-plasma-ZnS in unit cell is more useful for broad band filtering and other plasma functioning devices compared to the structure having glass-plasma-MgF₂ in one unit cell.     

Digital flow control of electroosmotic pump: Onsager coefficients and interfacial parameters determination

Electroosmosis and streaming potential are two complementary electrokinetic processes related by the Onsager relation. In particular, an electroosmotic pump (EOP) is potentially useful for a variety of engineering and bio-related applications. By fabricating samples consisting of dry-etched cylindrical pores (50 μm in length and 3.5 μm in diameter) on silicon wafers, we demonstrate that the use of digital control via voltage pulses can resolve the flow regulation and stability issues associated with the EOP, so that the intrinsic characteristics of the porous sample medium may be revealed. Through the consistency of the measured electroosmosis (EO) and the streaming potential (SP) coefficients as required by the Onsager relation, we deduce the zeta potential and the surface conductivity, both physical parameters pertaining to the liquid-solid interface.     

Processing discontinuous displacement fields by a spatio-temporal derivative technique

In this paper, a digital image correlation (DIC) method coupling cross-correlation with spatio-temporal differential techniques was proposed for assessing discontinuous displacement fields. The accuracy and robustness of the algorithm was assessed on a set of numerical tests by processing computer generated speckled-pattern images. Fracture mechanical tests in mode I were considered, in which both in-plane and out-of-plane rigid-body movements were taken into account. The ability for recovering the analytical asymptotic displacement field in mode I was analysed, and stress intensity factor, crack opening displacement and crack tip location were used as quantitative parameters for validation purposes. Throughout these tests, the results obtained with the proposed method were systematically compared to the ones from Aramis DIC-2D commercial code. Globally, the results computed from both methods are in good agreement with reference values. However, due to the high spatial resolution (point-wise characteristic), a better matching of the displacements in the neighbour of discontinuities could be obtained by the proposed method.     

Superhydrophobicity of silica nanoparticles modified with polystyrene

Polystyrene/silica nanoparticles were prepared by radical polymerization of silica nanoparticles possessing vinyl groups and styrene with benzoyl peroxide. The resulting vinyl silica nanoparticles, polystyrene/silica nanoparticles were characterized by means of Fourier transformation infrared spectroscopy, scanning electron microscopy and UV-vis absorption spectroscopy. The results indicated that polystyrene had been successfully grafted onto vinyl silica nanoparticles via covalent bond. The morphological structure of polystyrene/silica nanoparticles film, investigated by scanning electron microscopy, showed a characteristic rough structure. Surface wetting properties of the polystyrene/silica nanoparticles film were evaluated by measuring water contact angle and the sliding angle using a contact angle goniometer, which were measured to be 159° and 2°, respectively. The excellent superhydrophobic property enlarges potential applications of the superhydrophobic surfaces.