Discrete Gronwall Inequality Research Articles

A Crank–Nicolson-type finite-difference scheme and its algorithm implementation for a nonlinear partial integro-differential equation arising from viscoelasticity

A Crank–Nicolson-type finite-difference scheme is proposed and analyzed for a nonlinear partial integro-differential equation arising from viscoelasticity. The time derivative is approximated by the Crank–Nicolson scheme and the Riemann–Liouville fractional integral term is treated by means of the trapezoidal convolution quadrature rule. To construct a fully discrete difference scheme, the standard centered difference formula is utilized to approximate the second-order spatial derivative and the Galerkin method based on piecewise linear test functions is used to discrete the nonlinear convection term. Theoretical results of stability and convergence are derived using the non-negative character of the real quadratic form associated with the convolution quadrature, and combining with the discrete Gronwall inequality. Besides, a fixed point iterative numerical algorithm, which fills the gap that the existed numerical schemes have only theoretical analysis but no numerical results, is presented. Numerical results show the efficiency and feasibility of our scheme, and the orders of convergence are in good agreement with the theoretical results.

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time t=0. Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional diffusion problem is investigated. The error analysis is based on a nonstandard discrete Gronwall inequality. The final superconvergence result shows that an optimal grading of the temporal mesh should be selected as rgeq (2-alpha )/alpha . Numerical results confirm that our analysis is sharp.

Superconvergence Error Estimate of a Finite Element Method on Nonuniform Time Meshes for Reaction–Subdiffusion Equations

In this paper, we consider superconvergence error estimates of finite element method approximation of Caputo’s time fractional reaction–subdiffusion equations under nonuniform time meshes. For the standard Galerkin method we see that the optimal order error estimate of temporal direction cannot be derived from the weak formulation of the problem. We establish a time-space error splitting argument, which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully based on an improved discrete Gronwall inequality. We obtain the sharp temporal $$H^1$$ -norm error estimates with respect to the convergence order of the approximate solution and $$H^1$$ -norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global $$H^1$$ -norm superconvergence results are presented. Finally, we present some numerical results that give insight into the reliability of the theoretical analysis.

A domain decomposition method for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition

In this article a domain decomposition method is proposed to solve a time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition. Robin boundary conditions between the Navier-Stokes domain and Darcy domain are constructed by directly re-organizing the terms in the three interface conditions, including the Beavers-Joseph condition. In order to avoid the traditional iteration for the domain decomposition method at each time step, the interface information, which is needed for the Robin type transmission conditions at the current time step, is directly predicted based on the numerical solution of the previous time steps. Backward Euler scheme is first utilized for the temporal discretization while finite elements are used for the spatial discretization. The convergences of this domain decomposition method are rigorously analyzed for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition. The major difficulties in the analysis arise from nonlinear terms and Beavers-Joseph interface condition, including a series of technical treatments and the final special norm used in the discrete Gronwall's inequality for the analysis of full discretization. Based on the above preparation, we further develop a Lagrange multiplier method under the framework of the domain decomposition method to overcome the difficulty of non-unique solutions arising from the defective boundary condition. One interesting finding of this paper is that the Lagrange multipliers are time dependent functions instead of constants. In order to improve the accuracy order for the temporal discretization, a three-step backward differentiation scheme is used to replace the backward Euler scheme. Compared with the first scheme, the second one allows us to use the relative larger time step to reduce the computational cost while keeping the same accuracy. Numerical examples are provided to illustrate the features of the proposed method.

Finite-time stability of a class of nonlinear fractional delay difference systems

In this paper, we firstly develop an inequality on the rising function, which help us to deal with the fractional integral (sum) in terms of Hölder inequality. Secondly, a finite-time stability criterion of a class of nonlinear fractional delay difference systems with order 1<α<2 is addressed by virtue of a new discrete Gronwall inequality and Jensen inequality. Finally, two numerical examples are presented to demonstrate our theoretical result.

A novel discrete Gronwall inequality in the analysis of difference schemes for time-fractional multi-delayed diffusion equations

The theoretical analysis (convergence and stability) of an L2−1σ difference method is presented for nonlinear time-fractional diffusion equations with multiple delays. In this communication, we state and prove a new discrete form of a fundamental fractional Gronwall inequality. For the problems under consideration, that result and the L2−1σ formula are the cornerstones to the establishment of the optimal error estimates of our fully discrete linear difference scheme. Our numerical simulations confirm the validity of the theoretical findings. Moreover, both the analytical and the numerical results demonstrate superior rates of convergence when compared to similar methods of the literature.

An Effective Algorithm for Delay Fractional Convection-Diffusion Wave Equation Based on Reversible Exponential Recovery Method

In this paper, we investigate a linearized finite difference scheme for the variable coefficient semi-linear fractional convection-diffusion wave equation with delay. Based on reversible recovery technique, the original problems are transformed into an equivalent variable coefficient semi-linear fractional delay reaction-diffusion equation. Then, the temporal Caputo derivative is discreted by using $L_{1}$ approximation and the second-order spatial derivative is approximated by the centered finite difference scheme. The numerical solution can be obtained by an inverse exponential recovery method. By introducing a new weighted norm and applying discrete Gronwall inequality, the solvability, unconditionally stability, and convergence in the sense of $L_{2}$ - and $L_{\infty }$ - norms are proved rigorously. Finally, we present a numerical example to verify the effectiveness of our algorithm.

On the Numerical Approximability of Stable Dynamical Systems

This paper discusses the numerical approximation of stable dynamical systems of ordinary differential equations by general time-stepping methods. The traditional error analysis for this approximation yields estimates for the global discretization error in terms of the local truncations errors with constants generally growing exponentially in time due to the use of a discrete Gronwall inequality. In special cases of monotone systems, global error estimates can be shown to hold uniformly in time provided that also the time stepping scheme possesses certain monotonicity properties. The standard example is the backward Euler method. However, for more general time-stepping schemes, this is not clear even in the monotone case. It is shown here how in general situations certain qualitative stability properties (exponential and quasi-exponential stability) of the solution to be approximated can be used for extending local error estimates to global ones holding uniformly in time. Further, the approximate solutions are likewise stable and, in the autonomous case, tend to equilibrium points.

D-type iterative learning control for a class of parabolic partial difference systems

In this paper, an iterative learning control problem is addressed for a class of parabolic partial difference systems. Several discrete D-type iterative learning control algorithms with initial state learning are proposed for the systems which have no direct channel between the input and output as well as the initial state value being unfixed in the learning process. Based on fundamental mathematical analysis tools and the discrete Gronwall inequality, sufficient conditions for tracking error convergence in the iterative domain for open-loop, closed-loop and open-closed-loop iterative learning control are established and proven respectively. Numerical simulations verify the effectiveness of the theoretical results.

A Crank--Nicolson Finite Element Method and the Optimal Error Estimates for the Modified Time-Dependent Maxwell--Schrödinger Equations

In this paper, we study a Crank--Nicolson finite element method for the modified Maxwell--Schrodinger equations, which are deduced from the time-dependent Maxwell--Schrodinger equations under some assumptions. We present an (almost) unconditionally optimal error estimate for the numerical algorithm which only requires the time step $\Delta t < 1$. The key to our analysis is twofold. For one thing, we use some tricks to tackle the tough nonlinear terms without applying the inverse inequality to estimate the $L^{\infty}$ norm of the numerical solutions and get rid of certain restrictions like $\Delta t \leq C \, h^{\alpha}$ on the time step. For another, we employ a new technique to weaken the time step restriction required by the application of the discrete Gronwall inequality. Numerical tests are carried out to verify the theoretical results.

Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels

The cubic ``convolution spline'' method for first kind Volterra convolution integral equations was introduced in P.J. Davies and D.B. Duncan, $\mathit {Convolution\ spline\ approximations\ of\ Volterra\ integral\ equations}$, Journal of Integral Equations and Applications \textbf {26} (2014), 369--410. Here, we analyze its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function and a new error bound obtained from a particular B-spline quasi-interpolant.

Iterative learning control for discrete parabolic distributed parameter systems

In this paper, iterative learning control (ILC) technique is applied to a class of discrete parabolic distributed parameter systems described by partial difference equations. A P-type learning control law is established for the system. The ILC of discrete parabolic distributed parameter systems is more complex as 3D dynamics in the time, spatial and iterative domains are involved. To overcome this difficulty, discrete Green formula and analogues discrete Gronwall inequality as well as some other basic analytic techniques are utilized. With rigorous analysis, the proposed intelligent control scheme guarantees the convergence of the tracking error. A numerical example is given to illustrate the effectiveness of the proposed method.

The absolute stability for general discrete Lurie systems with superposition of multiple nonlinearities

In this paper, with the application of the concept of absolute stability about the certain variable, the necessary and sufficient conditions of absolute stability are obtained, which are about general discrete Lurie systems with superposition of nonlinearities. Some sufficient conditions are also obtained via the discrete Gronwall inequality. A numerical example illustrates the effectiveness of the proposed results.

Note on fractional difference Gronwall inequalities

This note is a reaction on a bunch of fractional inequalities that appeared in the last few years, and that are all based on what claims to be a fractional discrete Gronwall inequality. However, we show by a counterexample that this inequality is not correct. Stimulated by this, the main aim of this note is to propose new inequalities and illustrate the results on examples. Asymptotic properties of a solution of a linear equation are studied as well. Moreover, a brief discussion of other related results is given.

The Finite Difference Methods for Fractional Ordinary Differential Equations

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

Stabilizability for nonlinear difference controls systems with multiple delays

In this paper, we investigate the problem of stabilizability for nonlinear difference controls systems with multiple delays. We present a generalized discrete Gronwall's inequality for stabilizability analysis of this systems. Based on a new Gronwall's inequality, sufficient conditions for stabilizability of this systems are obtained. Numerical examples illustrate the results are given.

Extrapolation for solving a system of weakly singular nonlinear Volterra integral equations of the second kind

This article discusses an extrapolation method for solving a system of weakly singular nonlinear Volterra integral equations of the second kind. Based on a generalization of the discrete Gronwall inequality and Navot's quadrature rule, the modified trapeziform quadrature algorithm is presented. The iterative algorithm for solving the discrete system possesses a high accuracy order O(h 2+α). After the asymptotic expansion of errors is proved, we can obtain an approximation with a higher accuracy order using extrapolation. An a posteriori error estimation is provided. Some numerical results are presented to illustrate the efficiency of our methods.

A projected discrete Gronwall's inequality with sub-exponential growth

Along with the increasing interest in (h, k)-dichotomy, more attentions are paid to sub-exponential growth in research of asymptotic behaviours. In this paper, we generalize a projected discrete Gronwall's inequality given in [J. Differ. Equ. Appl. 10 (2004), 661–689] to a general one, which may include both terms of sub-exponential growth inside the summation and non-monotonic terms outside the summation. We demonstrate our results with concrete non-monotonic functions and sub-exponential functions. We apply our results to estimating bounded solutions of a non-linear difference equation with an (h, k)-dichotomy.

Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind

Based on a new generalization of discrete Gronwall inequality in [L. Tao, H. Yong, A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equality of the second kind, J. Math. Anal. Appl. 282 (2003) 56–62], Navot's quadrature rule for computing integrals with the end point singularity in [I. Navot, A further extension of Euler–Maclaurin summation formula, J. Math. Phys. 41 (1962) 155–184] and a transformation in [P. Baratella, A. Palamara Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004) 401–418], a new quadrature method for solving nonlinear weakly singular Volterra integral equations of the second kind is presented. The convergence of the approximation solution and the asymptotic expansion of the error are proved, so by means of the extrapolation technique we not only obtain a higher accuracy order of the approximation but also get a posteriori estimate of the error.

Stability of partial difference equations governing control gains in infinite-dimensional backstepping

We examine the stability properties of a class of LTV difference equations on an infinite-dimensional state space that arise in backstepping designs for parabolic PDEs. The nominal system matrix of the difference equation has a special structure: all of its powers have entries that are −1, 0, or 1, and all of the eigenvalues of the matrix are on the unit circle. The difference equation is driven by initial conditions, additive forcing, and a system matrix perturbation, all of which depend on problem data (for example, viscosity and reactivity in the case of a reaction–diffusion equation), and all of which go to zero as the discretization step in the backstepping design goes to zero. All of these observations, combined with the fact that the equation evolves only in a number of steps equal to the dimension of its state space, combined with the discrete Gronwall inequality, establish that the difference equation has bounded solutions. This, in turn, guarantees the existence of a state-feedback gain kernel in the backstepping control law. With this approach we greatly expand, relative to our previous results, the class of parabolic PDEs to which backstepping is applicable.