Spatial Periodic Boundary Conditions Research Articles

Double reduction order method based conservative compact schemes for the Rosenau equation

In this paper, fourth-order compact difference schemes are derived, analyzed and tested at length for both one- and two-dimensional Rosenau equations under the spatial periodic boundary conditions on the basis of the double reduction order method and bilinear compact operator. We prove that these schemes satisfy mass and energy conservation law via the help of the energy method. In addition, the uniquely solvable, unconditional convergence and stability are all obtained with the convergence order four in space and order two in time under the L∞-norm. Several numerical examples are presented to support the theoretical results.

Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic equations for vorticity. Probabilistic representations for solutions of these linear equations are given. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot--Savart-type law. We show that the approximation is divergent free and of first order. The results are extended to two-dimensional stochastic Navier-Stokes equations with additive noise, where, in particular, we prove the first mean-square convergence order of the vorticity approximation.

Monopoles and the Polyakov loop

We propose a new order parameter of a $Z_N$ symmetry in SU(N) gauge theories in $4$ dimensional Minkowski space-time, assuming spatial periodic boundary conditions. It is given by $Tr(P\exp(i\int_c A_{\mu}dx^{\mu}))$ where the spatial path $c$ is taken, for example, along $x_1$ axis. The parameter vanishes when the $Z_N$ symmetry is preserved. We calculate the contribution of QCD monopoles to the order parameter and show that when the monopoles condense $\langle\Phi\rangle\neq 0$, it vanishes, while it does not vanish when they do not condense. These calculations are performed using a monopole field $\Phi$ canonically quantized in a model of dual superconductor.

Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications

In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.

Nonlinear Rupture of Thin Micropolar Liquid Film Under a Magnetic Field

AbstractThis paper investigates the rupture problem of a thin micropolar liquid film under a magnetic field on a horizontal plate, using long-wave perturbation to resolve nonlinear evolution equations with a free film interface. The governing equation is resolved using a finite difference method as part of an initial value problem for spatial periodic boundary conditions. The effect of a micropolar liquid under a magnetic field on the nonlinear rupture mechanism is studied in terms of the micropolar parameter, R, the Hartmann constant, m and the initial disturbance amplitude, H0. Modeling results indicate that the R, m and H0 parameters strongly affect the film flow. Enhancing the micropolar and magnetic effects is found to delay the rupture time. In addition, the results show that the film rupture time increases as the values of initial disturbance magnitude decrease. The micropolar and magnetic parameters indeed play a significant role in the film flow on a horizontal plate. Moreover, the optimum conditions can be found to alter stability of the film flow by controlling the applied magnetic field.

Layer methods for stochastic Navier–Stokes equations using simplest characteristics

We propose and study a layer method for stochastic Navier–Stokes equations (SNSE) with spatial periodic boundary conditions and additive noise. The method is constructed using conditional probabilistic representations of solutions to SNSE and exploiting ideas of the weak sense numerical integration of stochastic differential equations. We prove some convergence results for the proposed method including its first mean-square order. Results of numerical experiments on two model problems are presented.

Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence

We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in R3. This model consists in a Navier–Stokes-type system for the mean flow u and two further partial differential equations: an equation for the frequency ω and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder Ω×]0,T[ (Ω⊂R3 cube, 0<T<+∞) under spatial periodic boundary conditions on ∂Ω×]0,T[ and initial conditions in Ω×{0}. We present an existence result for a weak solution {u,ω,k} to the problem under consideration, with ω, k obeying the inequalities c1+t≤1ω≤t+c2 and k1/2ω≥c3t1/2 (c1,c2,c3=const>0).

A Discontinuous Galerkin Time Domain Framework for Periodic Structures Subject to Oblique Excitation

A nodal Discontinuous Galerkin (DG) method is derived for the analysis of time-domain (TD) scattering from doubly periodic PEC/dielectric structures under oblique interrogation. Field transformations are employed to elaborate a formalism that is free from any issues with causality that are common when applying spatial periodic boundary conditions simultaneously with incident fields at arbitrary angles of incidence. An upwind numerical flux is derived for the transformed variables, which retains the same form as it does in the original Maxwell problem for domains without explicitly imposed periodicity. This, in conjunction with the amenability of the DG framework to non-conformal meshes, provides a natural means of accurately solving the first order TD Maxwell equations for a number of periodic systems of engineering interest. Results are presented that substantiate the accuracy and utility of our method.

Probabilistic Methods for the Incompressible Navier–Stokes Equations With Space Periodic Conditions

We propose and study a number of layer methods for Navier‒Stokes equations (NSEs) with spatial periodic boundary conditions. The methods are constructed using probabilistic representations of solutions to NSEs and exploiting ideas of the weak sense numerical integration of stochastic differential equations. Despite their probabilistic nature, the layer methods are nevertheless deterministic.

Probabilistic Methods for the Incompressible Navier–Stokes Equations With Space Periodic Conditions

We propose and study a number of layer methods for Navier‒Stokes equations (NSEs) with spatial periodic boundary conditions. The methods are constructed using probabilistic representations of solutions to NSEs and exploiting ideas of the weak sense numerical integration of stochastic differential equations. Despite their probabilistic nature, the layer methods are nevertheless deterministic.

Persistent energy flow for a stochastic wave equation model in nonequilibrium statistical mechanics

We consider a one-dimensional partial differential equation system modeling heat flow around a ring. The system includes a Klein-Gordon wave equation for a field satisfying spatial periodic boundary conditions, as well as Ornstein-Uhlenbeck stochastic differential equations with finite rank dissipation and stochastic driving terms modeling heat baths. There is an energy flow around the ring. In the case of a linear field with different (fixed) bath temperatures, the energy flow can persist even when the interaction with the baths is turned off. A simple example is given.

Quasi-periodic Solutions of Completely Resonant Wave Equations with Quasi-periodically forced Vibrations

In this paper, we provide the existence of quasi-periodic solutions with two frequencies for a class of completely resonant nonlinear wave equations with quasi-periodically forced vibrations under the spatial periodic boundary conditions. We consider the frequencies vector (? 1,? 2) near the linear system. The proofs are based on the Variational Lyapunov-Schmidt reduction and Linking Theorem.

Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$

We consider 1D completely resonant nonlinear wave equations of the type$v_{t t}$ - $v_{x x}$$ = -v^3 + \mathcal{O}(v^4)$ with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time.

A Numerical Investigation of Flow in a Channel Obstructed by an Array of Square Rods (2nd Report, Investigation of a Criterion for Judging the Uniqueness and Validity of Numerical Results)

Two-dimensional numerical computation has been performed for unsteady laminar flow. Spatial periodic boundary conditions were adopted in the streamwise direction and adequacy of its treatment was particularly studied. The following results were obtained. When the numerical result fluctuates with the computational domain size, various low frequency components are observed remarkably in the streamwise velocity of the calculated result. Compared with channel flow obstructed by single rod of the same size, intensities of velocity fluctuation in this study are relatively high, and their distribution throughout the flow field is rather uniform. Averaged kinematic energy of the velocity fluctuation is adopted as a key parameter, and a discrimination map of the validity of numerical results is presented with a time scale ratio between mean flow and velocity fluctuation, the rod pitch and the blockage ratio. The time scale ratio can be well correlated with a geometric parameter obtained from the flow configuration, and the discrimination map can be converted to a map described with the geometric parameter and the channel Reynolds number.

The dimension of attractor of the 2D g-Navier–Stokes equations

The g-Navier–Stokes equations in spatial dimension 2 were introduced by Roh as ∂ u ∂ t − ν Δ u + ( u ⋅ ∇ ) u + ∇ p = f , with the continuity equation ∇ ⋅ ( g u ) = 0 , where g is a suitable smooth real valued function. Roh proved the existence of global solutions and the global attractor, for the spatial periodic and Dirichlet boundary conditions. Roh also proved that the global attractor A g of the g-Navier–Stokes equations converges (in the sense of upper continuity) to the global attractor A 1 of the Navier–Stokes equations as g → 1 in the proper sense. In this paper, we will estimate the dimension of the global attractor A g , for the spatial periodic and Dirichlet boundary conditions. Then, we will see that the upper bounds for the dimension of the global attractors A g converge to the corresponding upper bounds for the global attractor A 1 as g → 1 in the proper sense.

Applicability of Spatial Periodic Boundary Conditions to Numerical Computation of Flow in a Channel Obstructed by an Array of Square Rods.

Two-dimensional numerical computation has been performed for unsteady laminar flow. Spatial periodic boundary conditions were adopted in the streamwise direction and adequacy of this treatment was particularly studied by changing the computational domain size. The following results were obtained. The Strouhal number and the lift coefficient amplitude may change with the domain size in the same flow conditions. Therefore, the simulation for one unit periodic flow field, a "unit cell", is valid for limited flow condition which depends on the Reynolds number, blockage ratio of the rod and its pitch. Uniqueness of the numerical results, which means calculated results remain unchanged with the domain size, is judged by change of the phase difference between neighboring cells. When the numerical results are unique, the Strouhal number, drag coefficient and the lift coefficient amplitude remarkably increase with the blockage ratio. The lift coefficient amplitude decreases with the increase of the rod pitch.

Rupture of Thin Power-Law Liquid Film on a Cylinder

The dynamic rupture process of a thin power-law type non-Newtonian liquid film on a cylinder has been analyzed by investigating the stability to finite amplitude disturbances. The dynamics of the liquid film is formulated using the balance equations including a body force term due to van der Waals attractions. The governing equation for the film thickness was solved by finite difference method as part of an initial value problem for spatial periodic boundary conditions. A decrease in the cylinder radius will induce a stronger lateral capillary force and thus will accelerate the rupture process. The influence of the power-law exponent on rupture is discussed.

RUPTURE OF THIN MICROPOLAR LIQUID FILM ON A CYLINDER

The dynamic rupture process of a thin liquid film on a cylinder has been analyzed by investigating the stability to finite amplitude disturbances. The dynamics of the liquid film is formulated using the balance equations including a body force term due to van der Waals attractions. The governing equation for the film thickness was solved by finite difference method as part of an initial value problem for spatial periodic boundary conditions. A decrease in the cylinder radius will induce a stronger lateral capillary force and thus will accelerate the rupture process. The rupture time increases with the material parameter, Δ.

Nonlinear theory of tear film rupture.

Nonlinear thin film rupture has been analyzed by investigating the stability of tear films to finite amplitude disturbances. The dynamics of the liquid film is formulated using the Navier-Stokes equations, including a body force term due to van der Waals attractions. The governing equation was solved by the finite difference method as part of an initial value problem for spatial periodic boundary conditions. The rupture of the tear film covering the cornea and the formation of dry spots is an important phenomenon in various pathological states associated with a dry eye.

Effect of Electrostatic Field on Film Rupture

Nonlinear thin film rupture has been analyzed by investigating the stability of films under the influence of a nonuniform electrostatic field to finite amplitude disturbances. The dynamics of the liquid film is formulated using the Navier-Stokes equations including a body force term due to van der Waals attractions. The effect of the electric field is included in the analysis only in the boundary condition at the liquid vapor interface. The governing equation was solved by finite difference method as part of an initial value problem for spatial periodic boundary conditions. The electric field stabilizes the film and increases the time to rupture when a long wavelength perturbation is introduced.