The quantum dual string wave functional in Yang-Mills theories

From any solution of the classical Yang-Mills equations, we define a string wave functional based on the Wilson loop integral. Its precise definition is given by replacing the string by a finite set of N points, and taking the limit N → ∞. We show that this functional satisfies the Schrödinger equation of the relativistic dual string to leading order in N. We speculate about the relevance of this object to the quantum problem.     

A strongly conservative finite element method for the coupling of Stokes and Darcy flow

We consider a model of coupled free and porous media flow governed by Stokes and Darcy equations with the Beavers–Joseph–Saffman interface condition. This model is discretized using divergence-conforming finite elements for the velocities in the whole domain. Discontinuous Galerkin techniques and mixed methods are used in the Stokes and Darcy subdomains, respectively. This discretization is strongly conservative in H^div(Ω) and we show convergence. Numerical results validate our findings and indicate optimal convergence orders.     

Operator symbols in the description of observable-state systems

For an observable-state system with finite degrees of freedom N topological properties of the kernels and symbols belonging to the considered operators are investigated. For the operators of $\text{L}$⁺($\text{S}$) kernels and symbols are distributions and for density matrices o? they are smooth functions.     

Application of group representation theory to derive Hermite interpolation polynomials on a triangle

Current methods used to devise sets of Hermite interpolation polynomials of minimal order that ensure C⁽ⁿ⁾ continuity across triangular element boundaries in two dimensions are not readily extensible to higher dimensions. The extension of such methods is especially difficult when the number of degrees of freedom afforded by data at points is different from the number of degrees of freedom determined by the coefficients of a complete polynomial basis to a particular order. This work introduces a formalism based on group representation theory that can accomplish this task in general. The method is introduced through the derivation of C⁽¹⁾ continuous Hermite polynomials that interpolate data at the three vertices of an equilateral triangular element. These interpolation polynomials are reported here for the first time. The polynomials derived here are compared to the standard polynomials defined in a right triangle by using the two sets of polynomials to solve the Laplace equation over finite elements. The methodology presented here is of use in higher dimensional elements when the complete polynomial degrees of freedom exceed the total C⁽ⁿ⁾ degrees of freedom at the nodes.     

New principles for string/membrane unification

The target space theory of the N = (2,1) heterotic string may be interpreted as a theory of gravity coupled to matter in either 1 + 1 or 2 + 1 dimensions. Among the target space theories in 1 + 1 dimensions are the bosonic, type II, and heterotic string world-sheet field theories in a physical gauge. The (2 + 1)-dimensional version describes a consistent quantum theory of supermembranes in 10 + 1 dimensions. The unifying framework for all of these vacua is a theory of (2 + 2)-dimensional self-dual geometries embedded in 10 + 2 dimensions. There are also indications that the N = (2,1) string describes the strong-coupling dynamics of compactifications of critical string theories to two dimensions, and may lead to insights about the fundamental degrees of freedom of the theory.     

Boson expansions and quantization of time-dependent self-consistent fields (I). Particle-hole excitations

Two concrete methods are presented for quantizing the time-dependent Hartree equations in terms of boson operators. The first is the well-known infinite boson expansion analogous to the Holstein-Primakoff representation of angular momentum operators. The second, a new development, consists of finite boson quadratic forms, and is analogous to the Schwinger representation of angular momenta. In each case, a physical boson subspace can easily be constructed within which the full fermion dynamics is exactly duplicated. It therefore follows that quantization of the time-dependent Hartree equations, including all degrees of freedom, retrieves the exact many-body problem. The discussion in this paper is limited to particle-hole excitations of an N-particle system. A generalization to one-nucleon transfer processes on the N-particle system is also given in terms of ideal odd nucleons, but this brings in infinite expansions.     

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.     

Adjoint-based h–p adaptive discontinuous Galerkin methods for the 2D compressible Euler equations

In this paper, we investigate and present an adaptive Discontinuous Galerkin algorithm driven by an adjoint-based error estimation technique for the inviscid compressible Euler equations. This approach requires the numerical approximations for the flow (i.e. primal) problem and the adjoint (i.e. dual) problem which corresponds to a particular simulation objective output of interest. The convergence of these two problems is accelerated by an hp-multigrid solver which makes use of an element Gauss–Seidel smoother on each level of the multigrid sequence. The error estimation of the output functional results in a spatial error distribution, which is used to drive an adaptive refinement strategy, which may include local mesh subdivision (h-refinement), local modification of discretization orders (p-enrichment) and the combination of both approaches known as hp-refinement. The selection between h- and p-refinement in the hp-adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values. Numerical results for the inviscid compressible flow over an idealized four-element airfoil geometry demonstrate that both pure h-refinement and pure p-enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach, but requiring fewer degrees of freedom. The proposed hp-adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom, and results in significant savings in computational cost. A high-speed flow test case is used to demonstrate the ability of the hp-refinement approach for capturing strong shocks or discontinuities while improving functional accuracy.     

Quantum-statistical approach to gross properties of peripheral collisions between heavy nuclei

Non-equilibrium quantum-statistical mechanics is applied to peripheral collisions between heavy nuclei (A?40) where a large number of degrees of freedom are involved during the process. By eliminating the relative motion from the explicit consideration, the transitions between different channels are determined by a Liouville equation with timedependent coupling matrix elements. The introduction of subsets of channels (coarse graining) leads to the definition of macroscopic variables which correspond to observable quantities. The equation of motion for the macroscopic variables become irreversible by assuming the values of the coupling matrix elements to be randomly distributed. The validity and possible applications of the resulting master equations are discussed.     

Dynamical approach to conformal gravity and the bosonic string effective action

We show that a theory invariant under the full local conformal group in a coset realization contains naturally the massless degrees of freedom of the closed bosonic string. This effective theory is alternative to the bosonic part of SupergravityN=1 as given by Manton and Chapline [4].     

Spectral element method in time for rapidly actuated systems

In this paper, the spectral element (SE) method is applied in time to find the entire time-periodic or transient solution of time-dependent differential equations. The time-periodic solution is computed by enforcing periodicity of the element set. Of particular interest are periodic forcing functions possessing high frequency content. To maintain the spectral accuracy for such forcing functions, an h-refinement scheme is employed near the semi-discontinuity without increasing the number of degrees of freedom.Time discretization by spectral elements is applied initially to a standard form of a set of linear, first-order differential equations subject to harmonic excitation and an excitation admitting rapid variation. Other case studies include the application of the SE approach to parabolic and hyperbolic partial differential equations. The first-order form of these equations is obtained through semi-discretization using conventional finite-element, spectral element and finite-difference schemes. Element clustering (h-refinement) is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation. The convergence in time of the method is demonstrated. In some cases, machine precision is obtained with 25 degrees of freedom per cycle. Finally the method is applied to a weakly nonlinear problem with time-periodic solution to demonstrate its future applicability to the analysis of limit-cycle oscillations in aeroelastic systems.     

An analysis of time discretization in the finite element solution of hyperbolic problems

The problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined. A modified equation analysis reveals that the classical, second-order accurate, time-stepping algorithms, i.e., the Lax-Wendroff, leap-frog, and Crank-Nicolson methods, properly combine with piecewise linear finite elements in advection problems only for small values of the time step. On the contrary, as the Courant number increases, the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit CFL property are not achieved. These fundamental numerical properties can, however, be recovered, while still remaining in the standard Galerkin finite element setting, by increasing the order of accuracy of the time discretization. This is accomplished by exploiting the Taylor series expansion in the time increment up to the third order before performing the Galerkin spatial discretization using piecewise linear interpolations. As a result, it appears that the proper finite element equivalents of second-order finite difference schemes are implicit methods of incremental type having third- and fourth-order global accuracy on uniform meshes (Taylor-Galerkin methods). Numerical results for several linear examples are presented to illustrate the properties of the Taylor-Galerkin schemes in one- and two-dimensional calculations.     

HCIZ integral and 2D Toda lattice hierarchy

The expression of the large N Harish Chandra–Itzykson–Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large N HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.     

Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element

A curve strip Fourier p-element for free vibration analysis of circular and annular sectorial thin plates is presented. The element transverse displacement is described by a fixed number of polynomial shape functions plus a variable number of trigonometric shape functions. The polynomial shape functions are used to describe the element's nodal displacements and the trigonometric shape functions are used to provide additional freedom to the edges and the interior of the element. With the additional Fourier degrees of freedom (dof) and reduce dimensions, the accuracy of the computed natural frequencies is greatly increased. Results are obtained for a number of circular and annular sectorial thin plates and comparisons are made with exact, the curve strip Fourier p-element, the proposed Fourier p-element and the finite strip element. The results clearly show that the curve strip Fourier p-element produces a much higher accuracy than the proposed Fourier p-element, the finite strip element.     

TIME INTEGRATION OF NON-LINEAR DYNAMIC EQUATIONS BY MEANS OF A DIRECT VARIATIONAL METHOD

Non-linear dynamic problems governed by ordinary (ODE) or partial differential equations (PDE) are herein approached by means of an alternative methodology. A generalized solution named WEM by the authors and previously developed for boundary value problems, is applied to linear and non-linear equations. A simple transformation after selecting an arbitrary interval of interest T allows using WEM in initial conditions problems and others with both initial and boundary conditions. When dealing with the time variable, the methodology may be seen as a time integration scheme. The application of WEM leads to arbitrary precision results. It is shown that it lacks neither numerical damping nor chaos which were found to be present with the application of some of the time integration schemes most commonly used in standard finite element codes (e.g., methods of central difference, Newmark, Wilson-θ, and so on). Illustrations include the solution of two non-linear ODEs which govern the dynamics of a single-degree-of-freedom (s.d.o.f.) system of a mass and a spring with two different non-linear laws (cubic and hyperbolic tangent respectively). The third example is the application of WEM to the dynamic problem of a beam with non-linear springs at its ends and subjected to a dynamic load varying both in space and time, even with discontinuities, governed by a PDE. To handle systems of non-linear equations iterative algorithms are employed. The convergence of the iteration is achieved by takingn partitions of T. However, the values of T/n are, in general, several times larger than the usual Δt in other time integration techniques. The maximum error (measured as a percentage of the energy) is calculated for the first example and it is shown that WEM yields an acceptable level of errors even when rather large time steps are used.     

Failsafe flux limiting and constrained data projections for equations of gas dynamics

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L² projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L² projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.     

Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics

We present hybridizable discontinuous Galerkin methods for solving steady and time-dependent partial differential equations (PDEs) in continuum mechanics. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; a judicious choice of the numerical flux to provide stability and consistency; and a global jump condition that enforces the continuity of the numerical flux to arrive at a global weak formulation in terms of the numerical trace. The HDG methods are fully implicit, high-order accurate and endowed with several unique features which distinguish themselves from other discontinuous Galerkin methods. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, they provide, for smooth viscous-dominated problems, approximations of all the variables which converge with the optimal order of k + 1 in the L²-norm. Third, they possess some superconvergence properties that allow us to define inexpensive element-by-element postprocessing procedures to compute a new approximate solution which may converge with higher order than the original solution. And fourth, they allow for a novel and systematic way for imposing boundary conditions for the total stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the methods. In addition, they possess other interesting properties for specific problems. Their approximate solution can be postprocessed to yield an exactly divergence-free and H(div)-conforming velocity field for incompressible flows. They do not exhibit volumetric locking for nearly incompressible solids. We provide extensive numerical results to illustrate their distinct characteristics and compare their performance with that of continuous Galerkin methods.