Stability Of Scheme Research Articles

Unconditionally strong energy stable scheme for Cahn–Hilliard equation with second‐order temporal accuracy

We propose an unconditionally gradient stable scheme for solving the Cahn–Hilliard equation with second‐order accuracy in time using the effective time‐step analysis. The conventional convex splitting scheme is one of the most well‐known methods for solving gradient flows, and it guarantees both energy stability and first‐order accuracy in time. Recently, there are some researches, which are extended to the second‐order accuracy; however, most of results provide the proof of energy stability for a modified (pseudo) energy or in a weak sense. In this paper, we prove the energy stability of the proposed method with respect to the Ginzburg–Landau free energy functional. Moreover, the unique solvability, mass conservation, and accuracy of the proposed scheme are also proven based on the convex splitting approach. The numerical experiments are presented to show convergence rate, mass conservation, energy stability, and phase separation are in good agreement with the theory.

An energy stable linear BDF2 scheme with variable time-steps for the molecular beam epitaxial model without slope selection

An implicit–explicit linear BDF2 scheme with variable steps is proposed for the molecular beam epitaxial model without slope selection. The fourth-order surface diffusion term is treated implicitly and the nonlinear term is approximated by a second order explicit extrapolation. A new type regularization term Aτ2Δh2D2ϕn (D2ϕn represents the variable-step BDF2 formula) is designed to ensure unconditional energy stability in long-time simulations. Under the time-step ratio constraint 0<τk/τk−1<4.864, the linear BDF2 method preserves the modified discrete energy dissipation law unconditionally if a proper large constant A is chosen. Furthermore, with the help of discrete orthogonal convolution kernels and the corresponding convolution Young inequalities, the L2 norm stability and convergence analysis of the linear BDF2 scheme for the MBE model are established. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed scheme.

On the convergence of an unconditionally stable numerical scheme for the Q-tensor flow based on the invariant quadratization method

We present a convergence analysis towards a numerical scheme designed for Q-tensor flows of nematic liquid crystals. This scheme is based on the Invariant Energy Quadratization method, which introduces an auxiliary variable to replace the original energy functional. In this work, we show that given an initial value with H2 regularity, we can obtain a uniform H2 estimate on the numerical approximations for the Q-tensor flows and then deduce the convergence to a strong solution of the parabolic-type Q-tensor equation. We also show that the limit of the auxiliary variable is equivalent to the original energy functional term in the strong sense.

New preconditioning and half-sweep accelerated overrelaxation solution for fractional differential equation

The present paper investigates the approximate solution of a one-dimensional linear space-fractional diffusion equation using a new preconditioning matrix to develop an efficient half-sweep accelerated overrelaxation iterative method. The proposed method utilizes unconditionally stable implicit finite difference schemes to formulate the discrete approximation equation to the problem. The formulation employs the Caputo fractional derivative to treat the space-fractional derivative in the problem. The paper's focus is to assess the improvement in terms of the convergence rate of the solution obtained by the proposed iterative method. The numerical experiment illustrates the superiority of the proposed method in terms of solution efficiency against one of the existing preconditioned methods, preconditioned accelerated overrelaxation and implicit Euler method. The proposed method reveals the ability to compute the solution with lesser iterations and faster computation time than the preconditioned accelerated overrelaxation and implicit Euler method. The method introduced in the paper, half-sweep preconditioned accelerated overrelaxation, has the potential to solve a variety of space-fractional diffusion models efficiently. Future investigation will improve the absolute errors of the solutions.

Automatic approach procedure of a flying vehicle on a mobile platform using backstepping controller

This paper presents the automatic approach procedure of a flying vehicle, attached to an ABB 7600 robot, and a mobile platform, attached to a Stewart platform. Due to a nonlinear dynamic behavior, it is necessary to implement complex control, stabilization and guidance schemes. The proposed solution for this system includes the development of an algorithm based on a backstepping control method, the controller design methodology being based on Lyapunov's stability theory. The proposed command law requires that the states are known, but it is also necessary to introduce a series of state estimators. Tracking a mobile platform is critical in surveillance, reconnaissance and tracking missions, with the control methodology defining a clear distinction between translational and rotational dynamics. The proposed algorithm is developed by separating two types of states involving an inverse kinematics, known as algebraic kinematics, in which the dynamic movements of the two pieces of equipment are used. The dynamics of the ABB 7600 robot involves a movement with seven degrees of freedom, while the Stewart platform can be used with a movement of six degrees of freedom. The proposed algorithm is implemented in both Matlab software and experimental testing. This paper provides results in terms of generating dynamics for both devices that can be used for simulating different scenarios of aerospace missions.

An Intrinsically Stable MPC Approach for Anti-Jackknifing Control of Tractor-Trailer Vehicles

Tractor-trailer vehicles are affected by jackknifing, a phenomenon that consists in the divergence of the trailer hitch angle and ultimately causes the vehicle to fold up. For the case of backward motions, in which jackknifing can occur at any speed, we present a control method that drives the vehicle along generic reference Cartesian trajectories while avoiding the divergence of the hitch angle. This is obtained thanks to a feedback control law that combines two actions: a tracking term, computed using input–output linearization, and a corrective term, generated via IS-MPC, an intrinsically stable MPC scheme which is effective for stable inversion of non-minimum phase systems. The successful performance of the proposed anti-jackknifing control is verified through simulations and experiments on a purposely built one-trailer prototype. To show the generality of the approach, we also apply and test the proposed method on a two-trailer vehicle.

A PSO-Based Approach for Optimal Allocation and Sizing of Resistive-Type SFCLs to Enhance the Transient Stability of Power Systems

Transient stability improvement of power systems in the event of short-circuit faults has always been an important issue in power systems analysis and studies. Resistive-type superconducting fault current limiters (RSFCL), owing to their capability in restricting fault currents, have been often taken into account as an efficient method to improve the transient stability of a power system. Regarding technical constraints as well as economic concerns, optimal allocation and sizing of RSFCLs in a power system play a crucial role in their efficient utilization. This paper aims to continue the authors’ previous work and enhance the transient stability of power systems by proposing an optimization approach for optimal sizing and the allocation of various candidate numbers of RSFCLs, as the most employed type of SFCL and the most efficient one in transient stability improvement. To solve the optimization problem, a PSO-based algorithm is solved in MATLAB through an objective function and related constraints. The efficacy of the proposed algorithm is evaluated by numerical studies on the IEEE 39-Bus New England system in various scenarios through the assessment of critical fault clearing time (CCT) as well as the generators rotor angle deviations as two crucial criteria for the transient stability of power systems. Simulating the optimization results in DIgSILENT Power Factory indicates an evident enhancement of the power system transient stability via employing optimized RSFCLs resulted from the proposed optimization algorithm. Moreover, the level of transient stability enhancement highly depends on the number of optimized RSFCLs employed in the power system. The results of this paper present a helpful guideline for power system planners to select an appropriate stability scheme based on RSFCLs besides other related technical and economic issues.

Optimal Control Approach for Robot System Using LQG Technique

A two-wheeled self-balancing robot system bases on the physical problem of an inverted pendulum. Stabilization of this type of mobile robot requires applying an active control approach. This paper proposes an efficient Linear Quadratic Gaussian (LQG) optimal control for the two-wheeled robot system. The LQG (a combination of a Kalman Filter (KF) and Linear Quadratic Regulator (LQR)) controller is designed to stabilize the robot while reducing the effect of the process and measurement noises on its performance. The LQG controller parameters (elements of state and control weighting matrices of the LQR and KF) are optimally tuned using the Particle Swarm Optimization (PSO) optimization method. The robot stabilization scheme is simulated utilizing MATLAB software to validate the proposed PSO-LQG controller system. The effectiveness of the proposed controller is validated based on the control criteria parameters, which are rise time, settling time, maximum overshoot, and steady-state error. The results prove that the proposed PSO-LQG controller can give very good movement performance in terms of both transient and steady-state responses.

Mathematical and numerical model for the malaria transmission: Euler method scheme for a malarial model

This research study has developed a mathematical model for malaria disease which is not only applicable for the case when the recovered humans return to the susceptible class, but also provides the directions for the case when the recovered humans also return to the infectious class. The model is simulated by using the Euler, Runge–Kutta-4 (RK-4), and nonstandard finite difference (NSFD) scheme. Firstly, the model is simulated by the Euler scheme and RK4 scheme and obtained graphical depiction for the endemic equilibrium as well as for the disease-free equilibrium (DFE). Then the mathematical model of malaria is simulated by an NSFD scheme and its graphical interpretation shows that it is suitable for all step sizes, i.e., it gives converging results even for very large step sizes. It is shown that the NSFD scheme is an unconditionally stable numerical scheme at a large step size. It is concluded that parameter [Formula: see text] is greater than unity in the disease manifestation of the landlord population in the long term and when the parameter [Formula: see text] is less than unity then the DFE is asymptotically stable.

Stable secure key distribution scheme via orthogonal polarizations and a joint source-channel model.

Optical secure key distribution (SKD) based on reciprocity has been the subject of increasing discussion, for its inherent information-theoretic safety and because there is less occupation of fiber channels. The combination of reciprocal polarization and broadband entropy sources has proven effective in increasing the rate of SKD. However, the stabilization of such systems suffers from the limited span of polarization states and inconsistent polarization detection. The specific causes are analyzed in principle. To solve this issue, we propose a strategy for extracting secure keys from orthogonal polarizations. Optical carriers with orthogonal polarizations at interactive parties are modulated by external random signals using polarization division multiplexing dual-parallel Mach-Zehnder modulators. After bidirectional transmission through a 10-km fiber channel, error-free SKD with a rate of 2.07 Gbit/s is experimentally realized. The high correlation coefficient of the extracted analog vectors can be maintained for over 30 min. The proposed method is a step toward the development of secure communication with high speed and feasibility.

Frequency-Domain Q-Compensated Reverse Time Migration Using a Stabilization Scheme

Seismic attenuation occurs during seismic wave propagation in a viscous medium, which will result in a poor image of subsurface structures. The attenuation compensation by directly amplifying the extrapolated wavefields may suffer from numerical instability because of the exponential compensation for seismic wavefields. To alleviate this issue, we have developed a stabilized frequency-domain Q-compensated reverse time migration (FQ-RTM). In the algorithm, we use a stabilized attenuation compensation operator, which includes both the stabilized amplitude compensation operator and the dispersion correction operator, for the seismic wavefield extrapolation. The dispersion correction operator is calculated based on the frequency-domain dispersion-only viscoacoustic wave equation, while the amplitude compensation operator is derived via a stabilized division of two propagation wavefields (the dispersion-only wavefield and the viscoacoustic wavefield). Benefiting from the stabilization scheme in the amplitude compensation, the amplification of the seismic noises is suppressed. The frequency-domain cross-correlation imaging condition is exploited to obtain the compensated image. The layered model experiments demonstrate the effectiveness and great compensation performance of our method. The BP gas model examples further verify its feasibility and stability. The field data applications indicate the practicability of the proposed method. The comparison between the acoustic and compensated results confirms that the proposed method is able to compensate for the seismic attenuation while suppressing the amplification of the high-frequency seismic noise.

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

Numerical analysis of energy stable weak Galerkin schemes for the Cahn–Hilliard equation

In this paper, we construct and analyze the energy-stable weak Galerkin schemes for the Cahn–Hilliard equation in the mixed form. The energy stability depends on the newly defined discrete energy functional. We propose two robust weak Galerkin schemes with extra stabilizing terms in time: the first-order conditionally energy stable scheme of which the stability does not depend on spatial mesh size, and the second-order unconditionally energy stable scheme. Energy stabilities and error estimates for the fully discrete schemes are obtained. Numerical experiments are provided to validate the theoretical results. A classical phase transition model is presented to show the energy stable property of the schemes.

Local maximum-entropy approximation based stabilization methods for the convection diffusion problems

Local Maximum Entropy (LME) approximation in the field of elliptic partial differential equations is more advanced compared to convection dominated flow problems. This paper presents a comparative study on the LME based meshfree stabilization methods for convection dominated convection-diffusion problems. LME approximants account for the disposal of the elements and direct imposition of boundary conditions, whereas the stabilization methods are meant to deal with the issues associated with the non-self-adjoint convective term. A new meshfree stabilization scheme, residual free bubbles based multiscale element free Galerkin method is also presented and compared with streamline upwind Petrov Galerkin meshfree method and variational multiscale element free Galerkin Method. The effect of different priors and radius of support which define the LME basis are studied to test their importance in the three methods with respect to standard benchmark problems. Cubic spline prior based LME is found to be enough to produce better stability and accuracy for all the stabilization methods when compared to quartic spline and Gaussian priors. LME based variational multiscale element free Galerkin method is observed to yield better stability and accuracy for all the priors. The convergence rate for the three stabilization methods is observed to be in good agreement with the classical FEM-SUPG.

Solution of space–time tempered fractional diffusion-wave equation using a high-order numerical method

At the present work, we attempt to obtain a high order numerical scheme for the solution of space–time fractional tempered diffusion-wave equation in one and two dimensional cases. For this purpose, we propose an unconditionally stable numerical scheme of order O(τ2) to approximate this equation in time direction. Then, we obtain the fully discrete scheme using the spectral element method in spatial directions. Well-posedness and error estimate of fully discrete scheme are given in details. Finally, to demonstrate the accuracy and efficiency of the proposed method in comparison with other schemes in the literature, the results of two test problems are presented.

A second-order unconditionally stable method for the anisotropic dendritic crystal growth model with an orientation-field

In this article, we develop a linear, unconditionally energy stable computational scheme for solving the dendritic crystal growth model with the orientational field. We apply the phase field model to describe the evolution of crystal with rotation. The model, which couples the heat equation and anisotropic Allen–Cahn type equation, is a complicated nonlinear system. The time integration is based on the second-order Crank–Nicolson method. The anisotropic coefficient is treated by using the invariant energy quadratization. We mathematically prove that the proposed method is unconditionally energy stable. The second-order spatial and temporal accuracy will be preserved for the numerical approximation. Various computational tests are performed to show the accuracy, stability, and efficiency of the proposed scheme.

On an Eigenvalue Property of Summation-By-Parts Operators

Summation-By-Parts (SBP) methods provide a systematic way of constructing provably stable numerical schemes. However, many proofs of convergence and accuracy rely on the assumption that the SBP operator possesses a particular eigenvalue property. In this note, three results pertaining to this property are proven. Firstly, the eigenvalue property does not hold for all nullspace consistent SBP operators. Secondly, this issue can be addressed without affecting the accuracy of the method by adding a specially designed, arbitrarily small perturbation term to the SBP operator. Thirdly, all pseudospectral methods satisfy the eigenvalue property.

Analytical Estimation of Power Losses in a Dual Active Bridge Converter Controlled with a Single-Phase Shift Switching Scheme

Micro-grid solutions around the world rely on the operation of DC/DC power conversion systems. The most commonly used solution for these topologies is the use of a dual active bridge (DAB) converter. Increasing the efficiency and reliability of this system contributes to the improvement in the stability of the entire microgrid. This paper discussed an analytical method of energy efficiency and power loss estimation in a single phase dual active bridge (DAB) converter controlled with a single-phase shift (SPS) modulation scheme for microgrid system stability. The presented approach uses conduction and commutation losses of semiconductors and high frequency transformer. All parameters required for the calculation may be obtained from the manufacturers’ datasheets or can be based on a simple measurement. The approach was validated by the comparison of the estimated energy efficiency characteristics with the measured ones for a prototype of a 5 kW single phase DAB converter equipped with silicon carbide metal-oxide semiconductor field-effect transistors (SiC MOSFET).

Stable and efficient time second-order difference schemes for fractional Klein–Gordon–Zakharov system

In this paper, we develop and analyze two stable and efficient time second-order difference schemes for space fractional Klein–Gordon–Zakharov (KGZ) system. The former is based on the multi-point weighted time second-order scheme to construct energy conservative linearized difference scheme. The latter is on the basis of time second-order splitting technique to develop accurate, efficient, linear and decoupled difference scheme. A main idea of the latter is to split the original fractional KGZ system into linear and nonlinear parts, and then to advance the subproblems with three stages. Furthermore, by utilizing the cut-off technique, energy analysis method and mathematical induction method, we obtain the error estimates of the former scheme, without any restrictive conditions on the grid ratio, compared with the restrictive conditions demanded in the existing literature. The feature of proposed schemes are linear decoupled, and easy to be applied in parallel computing, especially in long time simulations. At last, some numerical results are provided to validate the accuracy and efficiency of the proposed schemes, and simulate wave dynamics and interaction of the fractional KGZ system in one and two dimensions.

Advantageous symmetric cross-ownership and mergers

We model an industry where a subset of firms is interlinked via a reciprocal and symmetric share exchange agreement. Under quantity competition, a merger aiming at acquisition of market power can be reproduced by the same firms under a symmetric cross-ownership scheme. Under linear demand, a non-controlling symmetric cross-ownership scheme is always advantageous to its members if at least ( 2 − 2 ) ( 1 + n ) firms in an n -firm industry participate. The threshold drops to ( 1 + n ) / 2 for relatively low levels of cross-ownership. Cross-ownership schemes require fewer participants than mergers to be advantageous and can be more profitable than equal size mergers. Unlike mergers, a unique stable scheme exists. • Cournot mergers can be reproduced by symmetric and reciprocal cross-ownership. • Passive cross-ownership is unprofitable if less than half of industry participates. • Cross-ownership requires fewer participants than a merger to be advantageous. • Passive cross-ownership may be more profitable than an equal size merger. • Unlike the non-existence of stable mergers, a unique stable cross-ownership scheme exists.