Truncation Approximation Research Articles

Propagation of one-dimensional planar and nonplanar shock waves in nonideal radiating gas

The present study seeks to investigate a quasilinear hyperbolic system of partial differential equations which describes the unsteady one-dimensional motion of a shock wave of arbitrary strength propagating through a nonideal radiating gas. We have derived an infinite hierarchy of the transport equation which is based on the kinematics of one-dimensional motion of shock front. By using the truncation approximation method, an infinite hierarchy of transport equations, which governs the shock strength and the induced discontinuities behind it, is derived to study the kinematics of the shock front. The first three transport equations (i.e., first, second, and third-orders) are used to study the growth and decay behavior of shocks in van der Waals radiating gas. The decay laws for weak shock waves in nonradiating gas are entirely recovered in the second-order truncation approximation. The results obtained by the first three approximations for shock waves of arbitrary strength are compared with the results predicted by the characteristic rule. Also, the effect of nonideal parameters and radiation on the evolutionary behavior of shock waves are discussed and depicted pictorially.

Kinematics of spherical shock waves in an interstellar ideal gas clouds with dust particles

In this article, a system of partial differential equations governing the one‐dimensional motion of inviscid, self‐gravitating, spherical dusty gas cloud is considered. The evolutionary behavior of spherical shock waves of arbitrary strength in an interstellar dusty gas cloud is examined. By utilizing the method based on the kinematics of the one‐dimensional motion of shock waves, we derived an infinite set of transport equations governing the strength of shock waves and induced discontinuity behind it. By applying the truncation procedure to the infinite set of transport equations, we get an efficient system of finite number ordinary differential equations describing shock propagation, which can be regarded as a good approximation of the infinite hierarchy of the system. The truncated equations describing the shock strength and the induced discontinuity are used to analyze the growth and decay behavior of shock waves of arbitrary strength in the dusty gas medium. We considered the first two truncation approximations and the obtained results for the exponent from the successive approximation and compared our results with Guderley's exact similarity solution and the characteristic rule.

One-dimensional spherical shock waves in an interstellar dusty gas clouds

Abstract A system of partial differential equations describing the one-dimensional motion of an inviscid self-gravitating and spherical symmetric dusty gas cloud, is considered. Using the method of the kinematics of one-dimensional motion of shock waves, the evolution equation for the spherical shock wave of arbitrary strength in interstellar dusty gas clouds is derived. By applying first order truncation approximation procedure, an efficient system of ordinary differential equations describing shock propagation, which can be regarded as a good approximation of infinite hierarchy of the system. The truncated equations, which describe the shock strength and the induced discontinuity, are used to analyze the behavior of the shock wave of arbitrary strength in a medium of dusty gas. The results are obtained for the exponents from the successive approximation and compared with the results obtained by Guderley’s exact similarity solution and characteristic rule (CCW approximation). The effects of the parameters of the dusty gas and cooling-heating function on the shock strength are depicted graphically.

Modeliranje smanjenog reda i kontrola balansiranja biciklističkog robota

A new result for balancing control of a bicycle robot (bicyrobo), employing reduced-order modelling of a pre-specified design controller structure in higher-order to derive into a reduced controller has been presented in this paper. The bicyrobo, which is an unstable system accompanying other causes of uncertainty such as UN-model dynamics, parameter deviations, and external disruptions has been of great interests to researchers. The controllers in the literature reviews come up with the higher order controller (HOC), the overall system becomes complex from the perspective of analysis, synthesis, enhancement and also not easy to handle it's hardware implementation. Therefore, a reduced-order pre-specified controller is developed in this work. It is effective enough to tackle unpredictable dynamics. The reduced-order controller (ROC) design is based on model order reduction (MOR) method, which is a resutl of hybridization of balanced truncation (BT) and singular perturbation approximation (SPA) approach. The reduced model so obtained, which retains DC gain as well, has been named as balanced singular perturbation approximation (BSPA) approach. It is based upon the preservation of dominant modes (i.e. appropriate states) of the system as well as the removal of states having relatively less important distinguishing features. The strong demerit of the BT method is that, for reduced-order model (ROM), steady-state values or DC gain do not match with the actual system values. The BSPA has been enabled to account for this demerit. The method incorporates greater dominant requirements and contributes to a better approximation as compared to the existing methods. The results obtained by applying proposed controller, are compared with those of the controllers previously designed and published for the same type of work. Comparatively, the proposed controller has been shown to have better performance as HOC. The performance of HOC and ROC is also examined with perturbed bicyrobo in terms of time-domain analysis and performance indices error.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

Kinematics of one-dimensional spherical shock waves in interstellar van der Waals gas clouds

In this work, a system of non-linear partial differential equations, which describes one-dimensional motion of an inviscid, self-gravitating, and spherically symmetric van der Waals gas cloud, is considered. By using the method based on the kinematics of shock waves, the evolution equation for spherical shock wave in an interstellar van der Waals gas cloud is derived. By applying the truncation approximation procedure, an infinite system of transport equations, which governs the shock propagation, is derived to study the kinematics of shock waves for the one-dimensional motion. The first, second, and third order transport equations, which describe the shock strength and the induced discontinuity behind it, are used to analyze the decay and growth behavior of the shock waves in a non-ideal gas. The results are obtained for the exponent obtained from the first, second, and third order approximations and compared with the results obtained by Whitham’s characteristic rule (Chester–Chisnell–Whitham approximation). In addition, the effects of the parameters of non-idealness and cooling–heating function on the evolutionary behavior of shocks are discussed and shown graphically.

One-dimensional cylindrical shock waves in non-ideal gas under magnetic field

In the present paper, we analyze the evolutionary behavior of imploding strong shock waves propagating through a non-ideal gas in the presence of axial magnetic field. An evolution equation has been constructed by using the method based on the kinematics of one-dimensional motion of shock waves. The values of similarity exponents have been calculated by using the first order truncation approximation which describes the decay behavior of strong shocks. The approximate values of the similarity exponents are compared with the similarity exponents calculated by the CCW approximation, the exact similarity solution and perturbation technique.

A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions

We develop in this paper a Petrov-Galerkin spectral method for the inelastic Boltzmann equation in one dimension. Solutions to such equations typically exhibit heavy tails in the velocity space so that domain truncation or Fourier approximation would suffer from large truncation errors. Our method is based on the mapped Chebyshev functions on unbounded domains, hence requires no domain truncation. Furthermore, the test and trial function spaces are carefully chosen to obtain desired convergence and conservation properties. Through a series of examples, we demonstrate that the proposed method performs better than the Fourier spectral method and yields highly accurate results.

Modelling the penetration of magnetic flux in thin superconducting films with shell transformations

PurposeFinite element (FE) models are considered for the penetration of magnetic flux in type-II superconductor films. A shell transformation allows boundary conditions to be applied at infinity with no truncation approximation. This paper aims to determine the accuracy and efficiency of shell transformation techniques in such non-linear eddy current problems.Design/methodology/approachA three-dimensional H – ϕ formulation is considered, where the reaction field is calculated in the presence of a uniform applied field. The shell transformation is used in the far-field region, and the uniform applied field is introduced through surface terms, so as to avoid infinite energy terms. The resulting field distributions are compared against known solutions for different geometries (thin disks and thin strips in the critical state, square thin films). The influence of the shape, size and mesh quality of the far-field regions are discussed.FindingsThe formulation is shown to provide accurate results for a number of film geometries and shell transformation shapes. The size of the far-field region has to be chosen in such a way to properly capture the asymptotic decay of the fields, and a practical procedure to determine this size is provided.Originality/valueThe importance of the size of the far-field region in a shell transformation and its proximity to the conducting domains are both highlighted. This paper also provides a numerical way to apply a constant magnetic field in a given region, while the source, on which only the far-field behaviour of the applied field depends, is excluded from the model.

Finite-frequency power system reduction

In this paper, the finite-frequency model order reduction of large-scale power systems is studied. Two computationally efficient algorithms are presented for this purpose. The algorithms use the iterative rational Krylov subspace framework of the H2-model reduction and construct a reduced order model which ensures a superior approximation accuracy within the frequency region which contains the local, interarea, and/or interplant oscillations. The algorithms also preserve the slow and poorly damped poles of the original power system model by using a hybrid approach to combine the modal preservation property of modal truncation and superior approximation accuracy of moment matching. Accuracy within the desired frequency region is achieved by using the generalized Kalman-Yakubovich-Popov (GKYP) lemma-based frequency-dependent extended realizations. The performance of the proposed algorithms is tested on practical numerical examples and a comparison with the well-known techniques is presented to highlight the efficacy of the proposed algorithms.

Shock Wave Kinematics in a Relaxing Gas with Dust Particles

Abstract In this article, we considered the evolutionary behaviour of one-dimensional shock waves propagating through a relaxing gas with dust particles in a duct with spatially varying cross section. We adopted the procedure based on the kinematics of a one-dimensional motion to derive an infinite hierarchy of transport equations, which describe the evolutionary behaviour of shock of arbitrary strength propagating through the medium. The first three truncation approximations are considered, and the results are compared with existing results in the absence of relaxation and dust particles. The effects of dust particles and relaxation are studied using numerical computations. The results are depicted for different values of dust and relaxation parameters.

Controllable precision of the projective truncation approximation for Green’s functions**Project supported by the National Key Basic Research Program of China (Grant No. 2012CB921704), the National Natural Science Foundation of China (Grant No. 11374362), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 15XNLQ03).

Recently, we developed the projective truncation approximation for the equation of motion of two-time Green’s functions (Fan et al., Phys. Rev. B 97, 165140 (2018)). In that approximation, the precision of results depends on the selection of operator basis. Here, for three successively larger operator bases, we calculate the local static averages and the impurity density of states of the single-band Anderson impurity model. The results converge systematically towards those of numerical renormalization group as the basis size is enlarged. We also propose a quantitative gauge of the truncation error within this method and demonstrate its usefulness using the Hubbard-I basis. We thus confirm that the projective truncation approximation is a method of controllable precision for quantum many-body systems.

Model Boundary Approximation Method as a Unifying Framework for Balanced Truncation and Singular Perturbation Approximation

We show that two widely accepted model reduction techniques, balanced truncation (BT) and balanced singular perturbation approximation (BSPA), can be derived as limiting approximations of a carefully constructed parameterization of linear time invariant systems by employing the model boundary approximation method (MBAM) [1] . We also show that MBAM provides a novel way to interpolate between BT and BSPA, by exploring the set of approximations on the boundary of the “model manifold,” which is associated with the specific choice of model parameterization and initial condition and is embedded in a sample space of measured outputs, between the elements that correspond to the two model reduction techniques. This paper suggests similar types of approximations may be obtainable in topologically similar places (i.e., on certain boundaries) on the associated model manifold of nonlinear systems if analogous parameterizations can be achieved, therefore extending these widely accepted model reduction techniques to nonlinear systems. 1

The generalised singular perturbation approximation for bounded real and positive real control systems

The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well.

Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation

In this article, we study some soliton-type analytical solutions of Schrodinger equation, with their numerical treatment by Galerkin finite element method. First of all, some analytical solutions to the equation are obtained for different values of parameters; thereafter, the problem of truncating infinite domain to finite interval is taken up and truncation approximations are worked out for finding out appropriate intervals so that information is not lost while reducing the domain. The benefit of domain truncation is that we do not need to introduce artificial boundary conditions to find out numerical approximations. To verify theoretical results, numerical simulations are performed by Galerkin finite element method. Crank–Nicolson method is used for the time discretization, and non-linearity is resolved using predictor corrector method, which is second order accurate and computationally efficient.

Continuum Approximations to Systems of Correlated Interacting Particles

We consider a system of interacting particles with random initial conditions. Continuum approximations of the system, based on truncations of the BBGKY hierarchy, are described and simulated for various initial distributions and types of interaction. Specifically, we compare the Mean Field Approximation (MFA), the Kirkwood Superposition Approximation (KSA), and a recently developed truncation of the BBGKY hierarchy (the Truncation Approximation - TA). We show that KSA and TA perform more accurately than MFA in capturing approximate distributions (histograms) obtained from Monte Carlo simulations. Furthermore, TA is more numerically stable and less computationally expensive than KSA.

Synchronization of Phase Oscillators on the Hierarchical Lattice

Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction betweenthe oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

AbstractLetPbe the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vectorπT, and let(n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n+ 1) x (n+ 1) northwest corner truncation ofParbitrarily, with invariant probability vector(n)πT. We derive computableV-norm bounds on the error betweenπTand(n)πTin terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 asntends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.

Error bounds for augmented truncation approximations of continuous-time Markov chains

This paper considers the augmented truncation approximation of the generator of an ergodic continuous-time Markov chain with a countably infinite state space. The main purpose of this paper is to present bounds for the absolute difference between the stationary distributions of the original generator and its augmented truncation. As examples, we apply the bounds to an M∕M∕s retrial queue and an upper Hessenberg Markov chain.

Projective truncation approximation for equations of motion of two-time Green's functions

In the equation of motion approach to the two-time Green's functions, conventional Tyablikov-type truncation of the chain of equations is rather arbitrary and apt to violate the analytical structure of Green's functions. Here, we propose a practical way to truncate the equations of motion using operator projection. The partial projection approximation is introduced to evaluate the Liouville matrix. It guarantees the causality of Green's functions, fulfills the time translation invariance and the particle-hole symmetry, and is easy to implement in a computer. To benchmark this method, we study the Anderson impurity model using the operator basis at the level of Lacroix approximation. Improvement over conventional Lacroix approximation is observed. The distribution of Kondo screening in the energy space is studied using this method.