Time Convergence Rate Research Articles

Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases

We are concerned with the global existence and long time behavior of the solutions to the ES-FP model for diatomic gases proposed in [ 22 ]. The global existence of the solutions for this model near the global Maxwellian is established by nonlinear energy method based on the macro-micro decomposition. An algebraic convergence rate in time of the solutions to the equilibrium state is obtained by constructing the compensating function. Since the density distribution function \begin{document}$ F(t, x, v, I) $\end{document} also contains energy variable \begin{document}$ I $\end{document} , we derive more general Poincare inequality including variables \begin{document}$ v, I $\end{document} on \begin{document}$ \mathbb{R}^3\times \mathbb{R}^+ $\end{document} to establish the coercivity estimate of the linearized operator.

Finite-Time Control Algorithm Based on Modal State Observer for Flexible Satellite Attitude Tracking

An attitude controller with finite time convergence rate for satellite attitude tracking issue considering the flexible dynamic based on a finite time modal state observer is proposed in this article. The deformation of flexible appendages is taken into consideration and a finite-time state observer with auxiliary state to offset the initial error is proposed to get the flexible vibration estimation. The modal state and its derivative could be estimated by a full dimensional feedback observer. A finite time sliding mode based on the fraction order feedback of MRP is constructed and the finite time controller is proposed. The modal estimation is implied in controller design hence the counteraction of flexible system could be offset. System global finite-time stability is proved by Lyapunov method and the superiority is demonstrated by simulation results.

Robust Chattering-Free Finite Time Attitude Tracking Control with Input Saturation

his paper addresses the attitude tracking control problem of a rigid spacecraft in the presence of the modeling uncertainty, external disturbance, and saturated control input by designing two robust attitude tracking controllers. The basic controller is formulated using an integral sliding mode surface which is continuous and provides an asymptotic convergence rate for the closed-loop system. In this case, only the external disturbance with the prior information is considered. Then, to provide a finite time convergence rate and further improve the robustness of the control system under the unknown system uncertainty containing both the modeling uncertainty and external disturbance, a novel integral terminal sliding mode surface (ITSMS) is designed and associated with the continuous adaptive control method. Besides, a command filter is utilized to deal with the immeasurability problem within the proposed ITSMS and an auxiliary system to counteract the input saturation problem. Digital simulations are presented to verify the effectiveness of the proposed controllers.

L2-regularity result for solutions of backward doubly stochastic differential equations

We prove an [Formula: see text]-regularity result for the solutions of Forward Backward doubly stochastic differential equations (F-BDSDEs) under globally Lipschitz continuous assumptions on the coefficients. As an application of our result, we derive the rate of convergence in time for the (Euler time discretization-based) numerical scheme for F-BDSDEs proposed by Bachouch et al. (2016) under only globally Lipschitz continuous assumptions.

A stabilized phase-field method for two-phase flow at high Reynolds number and large density/viscosity ratio

Simulating two-phase flows in realistic industrial-complexity conditions remains an open problem. We present a phase-field method based on the Cahn-Hilliard equation that is able to simulate two-phase flow at high Reynolds number and at large density and viscosity ratios. We employ the entropy-viscosity method (EVM), applied both on the Navier-Stokes equations and phase-field equation, to stabilize the simulation in conjunction with an EVM-based artificial interface compression method (AICM) that maintains the sharpness of the interface. We implement this method based on a hybrid spectral-element/Fourier (SEF) discretization and demonstrate second-order accuracy in time and spectral convergence rate in space for smoothed fabricated solutions. We first test the accuracy and robustness of the stabilized SEF-EVM solver by solving the so-called three-dimensional “LeVeque problem” and compare against other available methods. Subsequently, we simulate a rising air bubble in a water container and find that the method is robust with respect to various parameters of the phase-field formulation. Lastly, we apply the method to simulate the onset and subsequent evolution of an air/oil slug in a long horizontal pipe using realistic parameters and incorporating gravity and surface tension effects. This is a particularly difficult flow to simulate with existing methods in realistic conditions and here we show that the new stabilized phase-field methods yields results in good agreement with the experimental data.

Pricing options under stochastic volatility jump model: A stable adaptive scheme

A non-uniform generation of the points for discretization of the spatial variables in pricing stochastic volatility jump models, such as the model of Bates, is given. The distribution attempts to concentrate on the hotzone at which the price of the option should be given with high precision. The approach is quadratically convergent in space and for the jump term, as well as with quartic rate of convergence in time. We show that the numerical procedure has conditional stability and is fast in pricing both American and European vanilla options under such models. In fact, enough criteria for global convergence of the discrete equations are discussed. Computational results for various options show the superiority and efficiency of our approach.

Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

This paper studies fully discrete approximations to the evolutionary Navier–Stokes equations by means of inf-sup stable $$H^1$$ -conforming mixed finite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound $$O(h^k)$$ for the $$L^2$$ error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method.

Energy and entropy preserving numerical approximations of thermodynamically consistent crystal growth models

We present a numerical scheme that preserves the total energy and the entropy production rate, termed the energy and entropy production rate preserving scheme, for a general class of thermodynamically consistent phase field models for dendritic crystal growth derived from the first and second law of thermodynamics. The scheme is second order in time, linear and energy and entropy production rate preserving for any time steps. The scheme is first discretized in time aided by the energy quadratization (EQ) method and then in space using compact finite difference methods. The linear system resulting from the scheme is shown to be uniquely solvable at both the semi-discrete and the fully discrete level. Mesh refinement tests are performed to show the second-order time convergence rate in the scheme. Several numerical examples of dendritic crystal growth are provided to demonstrate the accuracy and efficiency of the scheme. The effects of various model parameters on growth patterns of the crystal are further investigated in details with the numerical solver. The approach to developing the energy and entropy production rate-preserving numerical scheme proposed in this study is so general that it can be applied to a wide range of thermodynamically consistent models not limited to phase field models.

A combined finite volumes ‐ finite elements method for a low‐Mach model

SummaryIn this paper, we develop a finite volumes ‐ finite elements method based on a time splitting to simulate some low‐Mach flows. The mass conservation equation is solved by a vertex‐based finite volume scheme using a τ‐limiter. The momentum equation associated with the compressibility constraint is solved by a finite element projection scheme. The originality of the approach is twofold. First, the state equation linking the temperature, the density, and the thermodynamic pressure is imposed implicitly. Second, the proposed combined scheme preserves the constant states, in the same way as a similar one previously developed for the variable density Navier‐Stokes system. Some numerical tests are performed to exhibit the efficiency of the scheme. On the one hand, academic tests illustrate the ability of the scheme in term of convergence rates in time and space. On the other hand, our results are compared to some of the literature by simulating a transient injection flow as well as a natural convection flow in a cavity.

Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

A Second Order BDF Numerical Scheme with Variable Steps for the Cahn--Hilliard Equation

We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn--Hilliard equation. The construction relies on a second order backward difference, convex-splitting technique and viscous regularizing at the discrete level. We show that the scheme is unconditionally stable and uniquely solvable. In addition, under mild restriction on the ratio of adjacent time-steps, an optimal second order in time convergence rate is established. The proof involves a novel generalized discrete Gronwall-type inequality. As far as we know, this is the first rigorous proof of second order convergence for a variable step BDF2 scheme, even in the linear case, without severe restriction on the ratio of adjacent time-steps. Results of our numerical experiments corroborate our theoretical analysis.

Asymptotic stability of solutions to the Hamilton–Jacobi equation

This paper is concerned with the asymptotic behavior of the solution for the generalized Hamilton–Jacobi equation u t + f ( D u ) = Δ u . It is known that one dimensional equation v t + C 0 v x 2 = v x x has a self-similar solution v ( x 1 + t ) if v + ≠ v − where v + = v ( t , + ∞ ) , v − = v ( t , − ∞ ) . This kind of solution is called planar diffusion wave in multi-dimension (M-D). In this paper, it is shown that under some smallness conditions, the solutions to the generalized Hamilton–Jacobi equation converge to the above diffusion wave v ( x 1 + t ) with C 0 = 1 2 ∂ 2 f ( ξ 1 , ξ 2 , ⋯ , ξ n ) ∂ ξ 1 2 | ξ 1 = 0 , as time tends to infinity. The convergence rate in time is also obtained.

A quasilinear parabolic–elliptic chemotaxis-growth system with nonlinear secretion

ABSTRACTThis paper deals with the parabolic–elliptic chemotaxis-growth system with nonlinear secretion under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary, where , a>0, b>0, r>1, and is a smooth function satisfying with some and. It is shown, either or or then the system admits a unique global bounded classical solution. Moreover, we obtain the large time behavior of the solution for a specific logistic source. Finally, for the case and r=k+1, large time behavior and convergence rate are established.

Numerical analysis of a leapfrog ADI–FDTD method for Maxwell’s equations in lossy media

Recently, a so-called one-step leapfrog ADI–FDTD method has been developed in engineering community for solving the 3D time-dependent Maxwell’s equations. This method becomes quite popular in simulation wave propagation in graphene-based devices due to its efficiency. We investigate this method from a theoretical point of view by proving the energy conservation property, the unconditional stability of this ADI–FDTD method, and establishing the optimal second-order convergence rate in both time and space on non-uniform cubic grids. Numerical results are presented justifying our analysis.

Robust spacecraft attitude tracking control with integral terminal sliding mode surface considering input saturation

This paper studies the finite time spacecraft attitude tracking control problem, while considering modeling uncertainty, external disturbances and control input saturation. A novel integral terminal sliding mode surface (ITSMS) is designed by combining the fast terminal sliding mode surface (FTSMS) with a low pass filter to achieve a fast finite time convergence rate for the control system, without input singularity. An auxiliary signal is used to compensate for the effects of actuator saturation. The basic controller is first formulated based on the ITSMS, fast-TSM-type reaching law and auxiliary system, in the presence of an external disturbance and input saturation. Then, an adaptive control procedure is introduced, which simultaneously handles modeling uncertainty and external disturbance, thereby creating an adaptive attitude tracking controller. The proposed controller provides a fast finite time convergence rate for the control system, based on the newly designed ITSMS, while simultaneously compensating for modeling uncertainty, external disturbances and input saturation, without restricting the parameter selection process nor requiring repeated differentiation of nonlinear functions. Finally, digital simulation results are presented and demonstrate the effectiveness of the proposed controllers.

Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation

In this article, some second-order time discrete schemes covering parameter 𝜃 combined with Galerkin finite element (FE) method are proposed and analyzed for looking for the numerical solution of nonlinear cable equation with time fractional derivative. At time tk−𝜃, some second-order 𝜃 schemes combined with weighted and shifted Grunwald difference (WSGD) approximation of fractional derivative are considered to approximate the time direction, and the Galerkin FE method is used to discretize the space direction. The stability of second-order 𝜃 schemes is derived and the second-order time convergence rate in L2-norm is proved. Finally, some numerical calculations are implemented to indicate the feasibility and effectiveness for our schemes.

Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes–Darcy problem

We propose and study a combination of two second-order implicit–explicit (IMEX) methods for the coupled Stokes–Darcy system that governs flows in karst aquifers. The first is a second-order explicit two-step MacCormack scheme and the second is a second-order implicit Crank–Nicolson method. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. This combination so called the MacCormack rapid solver method is very efficient (faster, at least of second order accuracy in time and space) and can be easily implemented using legacy codes. Under time step limitation of the form Δt≤Ch (where h,Δt are mesh size and time step, respectively, and C is a physical parameter) we prove both long time stability and the rate of convergence of the method. Some numerical experiments are presented and discussed.

Global existence and long time behavior of the ellipsoidal-Fokker–Planck equation

ABSTRACTThe ellipsoidal-Fokker–Planck (ES-FP) model is proposed recently to obtain the correct Prandtl number in the numerical simulation of rarefied gas flows, which also satisfies the same properties as the Boltzmann equation, such as the H-theorem and the conservation laws of the mass, momentum, and energy. In this paper, we consider the Cauchy problem for this ES-FP equation in , and prove the global existence of the unique solution and establish its algebraic time convergence rate to the equilibrium state.

FINITE ELEMENT ALGORITHM BASED ON HIGH-ORDER TIME APPROXIMATION FOR TIME FRACTIONAL CONVECTION-DIFFUSION EQUATION

In this paper, finite element method with high-order approximation for time fractional derivative is considered and discussed to find the numerical solution of time fractional convection-diffusion equation. Some lemmas are introduced and proved, further the stability and error estimates are discussed and analyzed, respectively. The convergence result O(hr+1 + τ3-α) can be derived, which illustrates that time convergence rate is higher than the order (2-α) derived by L1-approximation. Finally, to validate our theoretical results, some computing data are provided.

A Markov chain Monte Carlo algorithm for Bayesian policy search

ABSTRACTPolicy search algorithms have facilitated application of Reinforcement Learning (RL) to dynamic systems, such as control of robots. Many policy search algorithms are based on the policy gradient, and thus may suffer from slow convergence or local optima complications. In this paper, we take a Bayesian approach to policy search under RL paradigm, for the problem of controlling a discrete time Markov decision process with continuous state and action spaces and with a multiplicative reward structure. For this purpose, we assume a prior over policy parameters and aim for the ‘posterior’ distribution where the ‘likelihood’ is the expected reward. We propound a Markov chain Monte Carlo algorithm as a method of generating samples for policy parameters from this posterior. The proposed algorithm is compared with certain well-known policy gradient-based RL methods and exhibits more appropriate performance in terms of time response and convergence rate, when applied to a nonlinear model of a Cart-Pole benchmark.