Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

A Novel Method to Enhance the Inversion Speed and Precision of the NMR T_(2) Spectrum by the TSVD Based Linearized Bregman Iteration

The low-field nuclear magnetic resonance(NMR)technique has been used to probe the pore size distribution and the fluid composition in geophysical prospecting and related fields.However,the speed and accuracy of the existing numerical inversion methods are still challenging due to the ill-posed nature of the first kind Fredholm integral equation and the contamination of the noises.This paper proposes a novel inversion algorithmto accelerate the convergence and enhance the precision using empirical truncated singular value decompositions(TSVD)and the linearized Bregman iteration.The L1 penalty term is applied to construct the objective function,and then the linearized Bregman iteration is utilized to obtain fast convergence.To reduce the complexity of the computation,empirical TSVD is proposed to compress the kernel matrix and determine the appropriate truncated position.This novel inversion method is validated using numerical simulations.The results indicate that the proposed novel method is significantly efficient and can achieve quick and effective data solutions with low signal-to-noise ratios.

Calculation of Electronic Structure of Anatase TiO_2 Doped with Transition Metal V,Cr,Fe and Cu Atoms by the Linearized Augmented Plane Wave Method

The electronic density of states and band structures of doped and un-doped anatase TiO2 were studied by the Linearized Augmented Plane Wave method based on the density functional theory. The calculation shows that the band structures of TiO2 crystals doped with transition metal atoms become narrower. Interesting, an excursion towards high energy level with increasing atomic number in the same element period could be observed after doping with transition metal atoms.

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.

MODELING OF NONLINEAR SYSTEMS BY MULTIPLE LINEARIZED MODELS

In order to design linear controller for nonlinear systems,a simple but efficient method of modeling a nonlinear system was proposed by means of multiple linearized models at different operating points in the entire range of the expected changes of the operating points.The original nonlinear system was described by linear combination of these multiple linearized models,with the linear combination parameters being identified on line based on least squares method.Model Predictive Control,an optimization based technique,was used to design the linear controller.A sufficient condition for ensuring the existence of a linear controller for the original nonlinear system was also given.Good performance indicated by two simulated examples confirms the usefulness of the proposed method.

Dynamic analysis on methanation reactor using a double-input–multi-output linearized model

A double-input–multi-output linearized system is developed using the state-space method for dynamic analysis of methanation process of coke oven gas.The stability of reactor alone and reactor with feed-effluent heat exchanger is compared through the dominant poles of the system transfer functions.With single or double disturbance of temperature and CO concentration at the reactor inlet,typical dynamic behavior in the reactor,including fast concentration response,slow temperature response and inverse response,is revealed for further understanding of the counteraction and synergy effects caused by simultaneous variation of concentration and temperature.Analysis results show that the stability of the reactor loop is more sensitive than that of reactor alone due to the positive heat feedback.Remarkably,with the decrease of heat exchange efficiency,the reactor system may display limit cycle behavior for a pair of complex conjugate poles across the imaginary axis.

A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation

To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation.

A Linearized and Unified Yield Criterion of Metals and Its Application

In this paper a linearized and unified yield crierion of metals is presented, which is in a form of a set of linear functions with two pararneters. The parameters are ex- pressed in terms of tension yield stress and so-called “shear-stretch ratio” and can bereadily determined from experimental data. It is shown that in stress space the set of yield functions is a set of polygons with twelve edges located between the Tresca’s hexagon and twin-shear-stress hexagon￣[1]. In this paper the present yield function isused to analyse the prestressiap loose running fit cylinders.

CAUCHY PROBLEM FOR LINEARIZED SYSTEM OF TWO-DIMENSIONAL ISENTROPIC FLOW WITH AXISYMMETRICAL INITIAL DATA IN GAS DYNAMICS

The explicit solution to Cauchy problem for linearized system of two-dimensional isentropic flow with axisymmetrical initial data in gas dynamics is given.

Steel Catenary Riser Fatigue Life Prediction Using Linearized Hydrodynamic Models

Steel catenary risers, (SCR) usually installed between seabed wellhead and floating platform are subjected to vortex shedding. These impose direct forces, hence cyclic stresses, and fatigue damage on the SCR. Riser failure has both economic and environmental consequences;hence the design life is usually greater than the field life, which is significantly reduced by vortex induced vibration (VIV). In this study, SCR and metOcean data from a field in Offshore Nigeria were substituted into linearized hydrodynamic models for simulations. The results showed that the hang off and touchdown regions were most susceptible to fatigue failure. Further analysis using Miner-Palm green models revealed that the fatigue life reduced from a design value of 20-years to 17.04-years, shortened by 2.96-years due to VIV. Furthermore, a maximum wave load of 5.154 kN was observed. The wave loads results corroborated with those obtained from finite element Orca Flex software, yielding a correlation coefficient of 0.975.

Application of Linearized Alternating Direction Multiplier Method in Dictionary Learning

The Alternating Direction Multiplier Method (ADMM) is widely used in various fields, and different variables are customized in the literature for different application scenarios [1] [2] [3] [4]. Among them, the linearized alternating direction multiplier method (LADMM) has received extensive attention because of its effectiveness and ease of implementation. This paper mainly discusses the application of ADMM in dictionary learning (non-convex problem). Many numerical experiments show that to achieve higher convergence accuracy, the convergence speed of ADMM is slower, especially near the optimal solution. Therefore, we introduce the linearized alternating direction multiplier method (LADMM) to accelerate the convergence speed of ADMM. Specifically, the problem is solved by linearizing the quadratic term of the subproblem, and the convergence of the algorithm is proved. Finally, there is a brief summary of the full text.

Linearized Equations of General Relativity and the Problem of Reduction to the Newton Theory

The paper is concerned with the problem of reduction of the general relativity theory to the Newton gravitation theory for a gravitation field with relatively low intensity. This problem is traditionally solved on the basis of linearized equations of general relativity which, being matched to the Newton theory equations, allow us to link the classical gravitation constant with the constant entering the general relativity equations. Analysis of the linearized general relativity equations shows that it can be done only for empty space in which the energy tensor is zero. In solids, the set of linearized general relativity equations is not consistent and is not reduced to the Newton theory equations. Specific features of the problem are demonstrated with the spherically symmetric static problem of general relativity which has the closed-form solution.

Breakage Distribution Estimation of Bauxite Based on Piecewise Linearized Breakage Rate

Laboratory tests were carried out to study the breakage kinetics of diasporic bauxite and determine its breakage distribution function. Non-first order breakage with different deceleration rates for different size intervals is found, which is most probably caused by the heterogeneity of the ore. Piecewise linearization method is proposed to describe the non-first order breakage according to its characteristics. In the method, grinding time is divided into several intervals and breakage is assumed to be first order in each interval. So, the breakage rates are calculated by taking the product of the last interval as feed and then established as a function of particle size and grinding time. Based on the predetermined breakage rate function, the breakage distribution of the ore is back-calculated from the experimental data using the population balance model (PBM). Finally, the obtained breakage parameters are validated and the simulated data are in good agreement with the experimental data. The obtained breakage distribution and the method for breakage rate description are both significant for modeling the full scale ball milling process of bauxite.

Prediction method of physical parameters based on linearized rock physics inversion

A linearized rock physics inversion method is proposed to deal with two important issues, rock physical model and inversion algorithm, which restrict the accuracy of rock physics inversion. In this method, first, the complex rock physics model is expanded into Taylor series to get the first-order approximate expression of the inverse problem of rock physics;then the damped least square method is used to solve the linearized rock physics inverse problem directly to get the analytical solution of the rock physics inverse problem. This method does not need global optimization or random sampling, but directly calculates the inverse operation, with high computational efficiency. The theoretical model analysis shows that the linearized rock physical model can be used to approximate the complex rock physics model. The application of actual logging data and seismic data shows that the linearized rock physics inversion method can obtain accurate physical parameters. This method is suitable for linear or slightly non-linear rock physics model, but may not be suitable for highly non-linear rock physics model.

The convergence properties of infeasible inexact proximal alternating linearized minimization

The proximal alternating linearized minimization(PALM)method suits well for solving blockstructured optimization problems,which are ubiquitous in real applications.In the cases where subproblems do not have closed-form solutions,e.g.,due to complex constraints,infeasible subsolvers are indispensable,giving rise to an infeasible inexact PALM(PALM-I).Numerous efforts have been devoted to analyzing the feasible PALM,while little attention has been paid to the PALM-I.The usage of the PALM-I thus lacks a theoretical guarantee.The essential difficulty of analysis consists in the objective value nonmonotonicity induced by the infeasibility.We study in the present work the convergence properties of the PALM-I.In particular,we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion.Based upon these,we manage to establish the stationarity of any accumulation point,and moreover,show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz property.The prominent advantages of the PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3-dimensional anisotropic frictional contact.

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work.It has the good property of discrete conservative energy.By the discrete energy analysis and mathematical induction method,it is proved to be uniquely solvable and unconditionally convergent with the secondorder accuracy in both time and space.Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law,unique solvability and unconditional convergence of order two in time and four in space.The resultant schemes preserve the maximum bound principle.The analysis techniques for convergence used in this paper also work for the Euler scheme,the Crank-Nicolson scheme and others.Numerical experiments are carried out to verify the computational efficiency,conservative law and the maximum bound principle of the proposed difference schemes.

Linearized microwave downconversion link based on fast and intelligent impairment equalization for noncooperative systems

We experimentally demonstrated the use of intelligent impairment equalization(IIE)for microwave downconversion link linearization in noncooperative systems.Such an equalizer is realized based on an artificial neural network(ANN).Once the training process is completed,the inverse link transfer function can be determined.With the inverse transformation for the detected signal after transmission,the third-order intermodulation distortion components are suppressed significantly without requiring any prior information from an input RF signal.Furthermore,fast training speed is achieved,since the configuration of ANN-based equalizer is simple.Experimental results show that the spurious-free dynamic range of the proposed link is improved to 106.5 dB·Hz^(2/3),which is 11.3 dB higher than that of a link without IIE.Meanwhile,the training epochs reduce to only five,which has the potential to meet the practical engineering requirement.

A Linearized Branch Flow Model Considering Line Shunts for Radial Distribution Systems and Its Application in Volt/VAr Control

When urban distribution systems are gradually modernized,the overhead lines are replaced by underground cables,whose shunt admittances can not be ignored.Traditional power flow(PF)model withπequivalent circuit shows non-convexity and long computing time,and most recently proposed linear PF models assume zero shunt elements.All of them are not suitable for fast calculation and optimization problems of modern distribution systems with non-negligible line shunts.Therefore,this paper proposes a linearized branch flow model considering line shunt(LBFS).The strength of LBFS lies in maintaining the linear structure and the convex nature after appropriately modeling theπequivalent circuit for network equipment like transformers.Simulation results show that the calculation accuracy in nodal voltage and branch current magnitudes is improved by considering shunt admittances.We show the application scope of LBFS by controlling the network voltages through a two-stage stochastic Volt/VAr control(VVC)problem with the uncertain active power output from renewable energy sources(RESs).Since LBFS results in a linear VVC program,the global solution is guaranteed.Case study exhibits that VVC framework can optimally dispatch the discrete control devices,viz.substation transformers and shunt capacitors,and also optimize the decision rules for real-time reactive power control of RES.Moreover,the computing efficiency is significantly improved compared with that of traditional VVC methods.

A review of linear friction welding of Ni -based superalloys

Ni-based superalloys are one of the most important materials employed in high-temperature applications within the aerospace and nuclear energy industries and in gas turbines due to their excellent corrosion,radiation,fatigue resistance,and high-temperature strength.Linear friction welding(LFW)is a new joining technology with near-net-forming characteristics that can be used for the manu-facture and repair of a wide range of aerospace components.This paper reviews published works on LFW of Ni-based superalloys with the aim of understanding the characteristics of frictional heat generation and extrusion deformation,microstructures,mechanical proper-ties,flash morphology,residual stresses,creep,and fatigue of Ni-based superalloy weldments produced with LFW to enable future optim-um utilization of the LFW process.

Influence of variable viscosity and double diffusion on the convective stability of a nanofluid flow in an inclined porous channel

The influence of variable viscosity and double diffusion on the convective stability of a nanofluid flow in an inclined porous channel is investigated.The DarcyBrinkman model is used to characterize the fluid flow dynamics in porous materials.The analytical solutions are obtained for the unidirectional and completely developed flow.Based on a normal mode analysis,the generalized eigenvalue problem under a perturbed state is solved.The eigenvalue problem is then solved by the spectral method.Finally,the critical Rayleigh number with the corresponding wavenumber is evaluated at the assigned values of the other flow-governing parameters.The results show that increasing the Darcy number,the Lewis number,the Dufour parameter,or the Soret parameter increases the stability of the system,whereas increasing the inclination angle of the channel destabilizes the flow.Besides,the flow is the most unstable when the channel is vertically oriented.