Dilations of Gamma-contractions by solving operator equations

For a contraction P and a bounded commutant S of P. we seek a solution X of the operator equation S - S*P = (1 - P* P)(1/2) X (1 - P* P)(1/2) where X is a bounded operator on (Ran) over bar (1 - P* P)(1/2) with numerical radius of X being not greater than 1. A pair of bounded operators (S, P) which has the domain Gamma = {(z(1) + z(2), z(2)): vertical bar z(1)vertical bar < 1, vertical bar z(2)vertical bar <= 1} subset of C-2 as a spectral set, is called a P-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Gamma-contraction (S, P). This allows us to construct an explicit Gamma-isometric dilation of a Gamma-contraction (S, P). We prove the other way too, i.e., for a commuting pair (S, P) with parallel to P parallel to <= 1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S, P) is a Gamma-contraction. We show that for a pure F-contraction (S, P), there is a bounded operator C with numerical radius not greater than 1, such that S = C + C* P. Any Gamma-isometry can be written in this form where P now is an isometry commuting with C and C. Any Gamma-unitary is of this form as well with P and C being commuting unitaries. Examples of Gamma-contractions on reproducing kernel Hilbert spaces and their Gamma-isometric dilations are discussed. (C) 2012 Elsevier Inc. All rights reserved.

A Numerical Scheme for Three-Dimensional Front Propagation and Control of Jordan Mode

As an example of a front propagation, we study the propagation of a three-dimensional nonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions and geometry of the wavefront are obtained by solving the conservation form of equations of a weakly nonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of seven conservation laws with an additional vector constraint, each of whose components is a divergence-free condition. This constraint is an involution for the system of conservation laws, and it is termed a geometric solenoidal constraint. The analysis of a Cauchy problem for the linearized system shows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode. For the numerical simulation of the conservation laws we employ a high resolution central scheme. The second order accuracy of the scheme is achieved by using MUSCL-type reconstructions and Runge-Kutta time discretizations. A constrained transport-type technique is used to enforce the geometric solenoidal constraint. The results of several numerical experiments are presented, which confirm the efficiency and robustness of the proposed numerical method and the control of the Jordan mode.

On the solution of bivariate population balance equations for aggregation: Pivotwise expansion of solution domain

The solution of a bivariate population balance equation (PBE) for aggregation of particles necessitates a large 2-d domain to be covered. A correspondingly large number of discretized equations for particle populations on pivots (representative sizes for bins) are solved, although at the end only a relatively small number of pivots are found to participate in the evolution process. In the present work, we initiate solution of the governing PBE on a small set of pivots that can represent the initial size distribution. New pivots are added to expand the computational domain in directions in which the evolving size distribution advances. A self-sufficient set of rules is developed to automate the addition of pivots, taken from an underlying X-grid formed by intersection of the lines of constant composition and constant particle mass. In order to test the robustness of the rule-set, simulations carried out with pivotwise expansion of X-grid are compared with those obtained using sufficiently large fixed X-grids for a number of composition independent and composition dependent aggregation kernels and initial conditions. The two techniques lead to identical predictions, with the former requiring only a fraction of the computational effort. The rule-set automatically reduces aggregation of particles of same composition to a 1-d problem. A midway change in the direction of expansion of domain, effected by the addition of particles of different mean composition, is captured correctly by the rule-set. The evolving shape of a computational domain carries with it the signature of the aggregation process, which can be insightful in complex and time dependent aggregation conditions. (c) 2012 Elsevier Ltd. All rights reserved.

An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.

SPLIT-STEP FORWARD MILSTEIN METHOD FOR STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper, we consider the problem of computing numerical solutions for stochastic differential equations (SDEs) of Ito form. A fully explicit method, the split-step forward Milstein (SSFM) method, is constructed for solving SDEs. It is proved that the SSFM method is convergent with strong order gamma = 1 in the mean-square sense. The analysis of stability shows that the mean-square stability properties of the method proposed in this paper are an improvement on the mean-square stability properties of the Milstein method and three stage Milstein methods.

Use of Fick's law and Maxwell–Stefan equations in computation of multicomponent diffusion

This article does not have an abstract.

Exponential compact higher order scheme and their stability analysis for unsteady convection-diffusion equations

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Peclet number 0 <= Pe <= 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.

Vorticity moments in four numerical simulations of the 3D Navier-Stokes equations

The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations on a periodic box 0, L](3) is addressed through four sets of numerical simulations that calculate a new set of variables defined by D-m(t) = (pi(-1)(0) Omega(m))(alpha m) for 1 <= m <= infinity where alpha(m) = 2m/(4m - 3) and Omega(m)(t)](2m) = L-3 integral(v) vertical bar omega vertical bar(2m) dV with pi(0) = vL(-2). All four simulations unexpectedly show that the D-m are ordered for m = 1,..., 9 such that Dm+1 < D-m. Moreover, the D-m squeeze together such that Dm+1/D-m NE arrow 1 as m increases. The values of D-1 lie far above the values of the rest of the D-m, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier-Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of 4096(3).

Solutions of a system of forced Burgers equation

In this article, we obtain explicit solutions of a system of forced Burgers equation subject to some classes of bounded and compactly supported initial data and also subject to certain unbounded initial data. In a series of papers, Rao and Yadav (2010) 1-3] obtained explicit solutions of a nonhomogeneous Burgers equation in one dimension subject to certain classes of bounded and unbounded initial data. Earlier Kloosterziel (1990) 4] represented the solution of an initial value problem for the heat equation, with initial data in L-2 (R-n, e(vertical bar x vertical bar 2/2)), as a series of self-similar solutions of the heat equation in R-n. Here we express the solutions of certain classes of Cauchy problems for a system of forced Burgers equation in terms of self-similar solutions of some linear partial differential equations. (C) 2013 Elsevier Inc. All rights reserved.

Structure-function hierarchies and von Karman-Howarth relations for turbulence in magnetohydrodynamical equations

We generalize the method of A. M. Polyakov, Phys. Rev. E 52, 6183 (1995)] for obtaining structure-function relations in turbulence in the stochastically forced Burgers equation, to develop structure-function hierarchies for turbulence in three models for magnetohydrodynamics (MHD). These are the Burgers analogs of MHD in one dimension Eur. Phys. J.B 9, 725 (1999)], and in three dimensions (3DMHD and 3D Hall MHD). Our study provides a convenient and unified scheme for the development of structure-function hierarchies for turbulence in a variety of coupled hydrodynamical equations. For turbulence in the three sets of MHD equations mentioned above, we obtain exact relations for third-order structure functions and their derivatives; these expressions are the analogs of the von Karman-Howarth relations for fluid turbulence. We compare our work with earlier studies of such relations in 3DMHD and 3D Hall MHD.

UNIQUENESS OF SOLUTIONS TO SCHRODINGER EQUATIONS ON H-TYPE GROUPS

This paper deals with the Schrodinger equation i partial derivative(s)u(z, t; s) - Lu(z, t; s) = 0; where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies vertical bar f(z, t)vertical bar less than or similar to q(alpha)(z, t), where q(s) is the heat kernel associated to L. If in addition vertical bar u(z, t; s(0))vertical bar less than or similar to q(beta)(z, t), for some s(0) is an element of R \textbackslash {0}, then we prove that u(z, t; s) = 0 for all s is an element of R whenever alpha beta < s(0)(2). This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rn+1.

Frequency Domain Based Solution for Certain Class of Wave Equations: An exhaustive study of Numerical Solutions

The paper discusses the frequency domain based solution for a certain class of wave equations such as: a second order partial differential equation in one variable with constant and varying coefficients (Cantilever beam) and a coupled second order partial differential equation in two variables with constant and varying coefficients (Timoshenko beam). The exact solution of the Cantilever beam with uniform and varying cross-section and the Timoshenko beam with uniform cross-section is available. However, the exact solution for Timoshenko beam with varying cross-section is not available. Laplace spectral methods are used to solve these problems exactly in frequency domain. The numerical solution in frequency domain is done by discretisation in space by approximating the unknown function using spectral functions like Chebyshev polynomials, Legendre polynomials and also Normal polynomials. Different numerical methods such as Galerkin Method, Petrov- Galerkin method, Method of moments and Collocation method or the Pseudo-spectral method in frequency domain are studied and compared with the available exact solution. An approximate solution is also obtained for the Timoshenko beam with varying cross-section using Laplace Spectral Element Method (LSEM). The group speeds are computed exactly for the Cantilever beam and Timoshenko beam with uniform cross-section and is compared with the group speeds obtained numerically. The shear mode and the bending modes of the Timoshenko beam with uniform cross-section are separated numerically by applying a modulated pulse as the shear force and the corresponding group speeds for varying taper parameter in are obtained numerically by varying the frequency of the input pulse. An approximate expression for calculating group speeds corresponding to the shear mode and the bending mode, and also the cut-off frequency is obtained. Finally, we show that the cut-off frequency disappears for large in, for epsilon > 0 and increases for large in, for epsilon < 0.

Modified Adomian Decomposition Method for Van der Pol equations

In the paper, the well known Adomian Decomposition Method (ADM) is modified to solve the parabolic equations. The present method is quite different than the numerical method. The results are compared with the existing exact or analytical method. The already known existing Adomian Decomposition Method is modified to improve the accuracy and convergence. Thus, the modified method is named as Modified Adomian Decomposition Method (MADM). The Modified Adomian Decomposition Method results are found to converge very quickly and are more accurate compared to ADM and numerical methods. MADM is quite efficient and is practically well suited for use in these problems. Several examples are given to check the reliability of the present method. Modified Adomian Decomposition Method is a non-numerical method which can be adapted for solving parabolic equations. In the current paper, the principle of the decomposition method is described, and its advantages are shown in the form of parabolic equations. (C) 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

The periodic 3D Navier-Stokes equations are analyzed in terms of dimensionless, scaled, L-2m-norms of vorticity D-m (1 <= m <= infinity). The first in this hierarchy, D-1, is the global enstrophy. Three regimes naturally occur in the D-1-D-m plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime 1]; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

Transition from dissipative to conservative dynamics in equations of hydrodynamics

We show, by using direct numerical simulations and theory, how, by increasing the order of dissipativity (alpha) in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case alpha -> infinity U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of alpha greater than a crossover value alpha(crossover). We thus provide a self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows.