An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations

We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier–Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure–velocity formulation of the incompressible Navier–Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k?0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H (div)-conforming, and converges with order k + 2 for k ? 1 and with order 1 for k = 0 in the L²-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.     

A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows

We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss–Lobatto–Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge–Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re = 400–5000 and flows around an impulsively started cylinder for Re = 550–9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.     

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and noneconvex Hamiltonian, optimal (k + 1)th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and have the solution converges to the viscosity solution.     

Two-phase electrohydrodynamic simulations using a volume-of-fluid approach: A comment

This comment refers to the article of Tomar et al. [1], which presents a numerical methodology of a continuum surface force formulation for simulating two-phase electrohydrodynamic flows. The present work shows, that due to the diffusive character of the Laplacian equation (? · (??₀E) = 0) with discontinuous physical properties (?(x, y, z)), different averaging methods (arithmetic and harmonic) for the fluid property in the transition region have to be applied. The correct choice of the averaging method depends on the orientation of the flux to the interface.An additional improvement is made by calculating the electric displacement D at the cell faces. This leads to a numerical solution independent of the spatial resolution as well as of the interfacial smearing. Simulation results of two different test cases show that the error of the numerical solution is in the order of machine precision.     

High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics

We present a class of hybridizable discontinuous Galerkin (HDG) methods for the numerical simulation of wave phenomena in acoustics and elastodynamics. The methods are fully implicit and high-order accurate in both space and time, yet computationally attractive owing to their following distinctive features. First, they reduce the globally coupled unknowns to the approximate trace of the velocity, which is defined on the element faces and single-valued, thereby leading to a significant saving in the computational cost. In addition, all the approximate variables (including the approximate velocity and gradient) converge with the optimal order of k + 1 in the L²-norm, when polynomials of degree k ? 0 are used to represent the numerical solution and when the time-stepping method is accurate with order k + 1. When the time-stepping method is of order k + 2, superconvergence properties allows us, by means of local postprocessing, to obtain better, yet inexpensive approximations of the displacement and velocity at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k + 2 in the L²-norm when k ? 1 for both acoustics and elastodynamics. Extensive numerical results are provided to illustrate these distinctive features.     

A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions

Standard Galerkin discretization techniques (with locally- or globally-supported basis functions) for boundary integral equations are inefficient for high frequency three dimensional exterior scattering simulations because they require a fixed number of unknowns per wavelength in each dimension, leading to large CPU time and memory requirements to set up the dense Galerkin matrix, with each entry requiring evaluation of multi-dimensional highly oscillatory integrals. In this work, using globally-supported basis functions, we describe an efficient fully discrete Galerkin surface integral equation algorithm for simulating high frequency acoustic scattering by three dimensional convex obstacles that includes a powerful integration scheme for evaluation of four dimensional Galerkin integrals with high-order accuracy. Such high-order order accuracy for various practically relevant frequencies (k ∈ [1, 100,000]) substantially improves on approximations based on standard asymptotic techniques. We demonstrate the efficiency of our algorithm for spherical and non-spherical convex scattering for several wavenumbers 1 ? k ? 100,000 for low to high order prescribed tolerance. Our fully discrete algorithm requires only mild growth in the number of unknowns and CPU time as the frequency increases.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

Compact optimal quadratic spline collocation methods for the Helmholtz equation

Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact; that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N² log N) on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition.     

Optimal Runge–Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge–Kutta time integrators, with the aim of deriving optimal Runge–Kutta schemes for wave propagation applications. We review relevant Runge–Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order Runge–Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge–Kutta methods, we provide the coefficients for a 2N-storage implementation, along with the information needed by the user to employ them optimally.     

An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection–diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton–Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is (p+1)$(p+1)$th order accurate and when polynomials of degree p?0$p?0$ are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p+1$p+1$ in the L²$L^2$-norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p+1$p+1$ in the L²$L^2$-norm. The new approximate scalar variable is shown to converge with order p+2$p+2$ in the L²$L^2$-norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.     

Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations

We present two hybridizable discontinuous Galerkin (HDG) methods for the numerical solution of the time-harmonic Maxwell’s equations. The first HDG method explicitly enforces the divergence-free condition and thus necessitates the introduction of a Lagrange multiplier. It produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the Lagrange multiplier. The second HDG method does not explicitly enforce the divergence-free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only. For both HDG methods, the approximate vector field converges with the optimal order of k + 1 in the L²-norm, when polynomials of degree k are used to represent all the approximate variables. We propose elementwise postprocessing to obtain a new H^curl-conforming approximate vector field which converges with order k + 1 in the H^curl-norm. We present extensive numerical examples to demonstrate and compare the performance of the HDG methods.     

A high order Discontinuous Galerkin – Fourier incompressible 3D Navier–Stokes solver with rotating sliding meshes

We present the development of a sliding mesh capability for an unsteady high order (order ? 3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier–Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular–quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian–Eulerian form of the incompressible Navier–Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x–y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier–Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier–Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.     

An inverse Sturm–Liouville problem with a fractional derivative

In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm–Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order α ∈ (1, 2) of fractional derivative is sufficiently away from 2.     

Propagation characteristics of Gaussian beams through 2 × 2 square matrix circular apertures

Based on the expansion of the hard aperture function into a finite sum of complex Gaussian functions, using Collins formula, the approximate analytical expression of Gaussian beams through 2 × 2 square matrix circular apertures is derived. Finally, some numerical examples are given to illustrate the analytical results by Matlab. It is shown that the propagation characteristics is related with the propagation distance z, the radius of circular aperture a and the distance d. The results can be directly used in other beams and square matrix apertures, and be applied to control beams and optical system designing.     

Characteristic Brewster angles for anisotropic interfaces

The effect of nanoscopically stratified anisotropic transmission layer between two isotropic dielectric medium on the characteristic Brewster angles is carried out. The analysis is based on the second-order approximate formulas for the quasi-Brewster, polarization (second Brewster), and principal (third Brewster) angles obtained in the framework of a long-wavelength approximation. The accuracy of these analytical expressions is estimated by using exact numerical methods for the solution of the anisotropic reflection problem. It is shown that the characteristics Brewster angles for real physical interfaces may differ considerably from the classical Brewster angle ϕ_B = arctg(n_b/n_a), for certain material parameters even then, if the thickness of a transition layer is very significantly smaller than the wavelength of optical radiation.     

The miscibility of CuxAg1 − xI using a Tersoff potential

Simulation methods have been used to study the miscibility ofCu_xAg_1 − xI based on a Tersoff potential. Monte Carlo calculations show that Cu_xAg_1 − xI is a complete solid solution. This result agrees well with experiments using NMR and X-ray diffractions methods. Structural, elastic and thermodynamic properties are also predicted at 0.25, 0.5 and 0.75 using molecular dynamics simulations.     

A linear stability analysis for nonlinear,grey, thermal radiative transfer problems

We present a new linear stability analysis of three time discretizations and Monte Carlo interpretations of the nonlinear, grey thermal radiative transfer (TRT) equations: the widely used “Implicit Monte Carlo” (IMC) equations, the Carter Forest (CF) equations, and the Ahrens–Larsen or “Semi-Analog Monte Carlo” (SMC) equations. Using a spatial Fourier analysis of the 1-D Implicit Monte Carlo (IMC) equations that are linearized about an equilibrium solution, we show that the IMC equations are unconditionally stable (undamped perturbations do not exist) if α, the IMC time-discretization parameter, satisfies 0.5 < α ? 1. This is consistent with conventional wisdom. However, we also show that for sufficiently large time steps, unphysical damped oscillations can exist that correspond to the lowest-frequency Fourier modes. After numerically confirming this result, we develop a method to assess the stability of any time discretization of the 0-D, nonlinear, grey, thermal radiative transfer problem. Subsequent analyses of the CF and SMC methods then demonstrate that the CF method is unconditionally stable and monotonic, but the SMC method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium. This stability theory provides new conditions on the time step to guarantee monotonicity of the IMC solution, although they are likely too conservative to be used in practice. Theoretical predictions are tested and confirmed with numerical experiments.     

Broadband super-collimation with low-symmetric photonic crystal

We investigate dispersive properties of two dimensional photonic crystal (PC) called star-shaped PC (STAR-PC) in order to succeed super-collimation over a broad bandwidth. Both time- and frequency-domain numerical methods are conducted. Due to introduced low-symmetry in the primitive cell, flat contours are observed at the fifth band for transverse magnetic mode. The proposed structure supports a super-collimation effect over a broad wavelength range between 1443 nm and 1701 nm with a bandwidth of Δω = 16.42%. The intrinsic characteristic of STAR-PC provides in-plane beam propagation with a limited diffraction length of 120a, where a is the lattice constant. By means of STAR-PC, one may realize super-collimation based single-mode optical devices with a low insertion loss, reduced dispersion and wide bandwidth.     

Electrical and structural properties of stepped,partially reconstructed Au(11n) surfaces in HClO4 and H2SO4 electrolytes

We have studied the interface capacitance and the voltammograms of Au(11n) (n = 5, 7, 11, 17) and of Au(100) electrodes in 5 mM HClO₄ and 5 mM H₂SO₄ after immersion into the electrolyte solution at ?0.4 V versus a saturated calomel electrode. The minima of the capacitance curves measured in positive sweeps continuously shift towards positive potentials as function of 1/n. All voltammograms, even that of Au(1 1 5), display peaks that are characteristic for lifting of surface reconstructions, albeit at different potentials. Thus, all vicinal surfaces appear to have at least sections that allow reconstruction. This inference is consistent with STM-profiles of an Au(1 1 9) surface which displays a wide range of local inclination angles corresponding to local (11n)-orientations with 3.5 < n < ∞. A numerical analysis of the voltammograms shows the existence of three different ranges of transition potentials for the lifting of the reconstruction. The transition potentials are assigned to three different structures of the reconstructed phase as either observed by experiment or proposed by theory.     

Conductivity of SnP2O7 and In-doped SnP2O7 prepared by an aqueous solution method

SnP₂O₇ and In-doped SnP₂O₇ have been prepared by an aqueous solution method using (NH₄)₂HPO₄ as phosphorous source. It was found that the solid solution limit in Sn_1 ? xIn_x(P₂O₇)_1 ? δ was at least x = 0.12. All pyrophosphates in the Sn_1 ? xIn_x(P₂O₇)_1 ? δ (x ≤ 0.12) series exhibit 3 × 3 × 3 superlattice structures. The conductivities of Sn_0.92In_0.08(P₂O₇)_1 ? δ in air are 6.5 × 10^? 6 and 8.0 × 10^? 9 S/cm at 900 and 400 °C, respectively, when prepared by an aqueous solution method and annealed at 1000 °C. The conductivity of undoped SnP₂O₇ is slightly lower. However, it was also found that the low-temperature conductivities of pyrophosphates annealed only at 650 °C are several orders of magnitude higher than those annealed at 1000 °C, which could be related to a trace amount of an amorphous secondary phase. The peak conductivity was in this case observed at around 250 °C, which is the same temperature as previously observed in In-doped SnP₂O₇ although the conductivity is still three orders of magnitude lower in the present study. These differences can be related to large differences in particle size and morphology, and all in all, the conductivities of SnP₂O₇-based materials are very sensitive to the synthetic history.