Numerical solutions to a BBM‐Burgers model with a nonlocal viscous term

In this paper, we numerically investigate the BBM‐Burgers equation with a nonlocal viscous term (1) where is the Riemann‐Liouville half derivative. In particular, we implement different numerical schemes to approximate the solution and its asymptotical behavior. Also, we compare our numerical results with those given in 2013, 2014 for similar models.     

New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

In this article, some high‐order accurate difference schemes of dispersive shallow water waves with Rosenau‐KdV‐RLW‐equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.     

Stability analysis for fractional‐order partial differential equations by means of space spectral time Adams‐Bashforth Moulton method

In this article, a new numerical scheme space Spectral time Fractional Adam Bashforth Moulton method for the solution of fractional partial differential equations is offered. The proposed method is obtained by modifying, in a suitable way; the spectral technique and the method of lines. The attention is focused on the stability properties and hence an elegant stability analysis for the current approach is also provided. Finally, two examples are presented to illustrate the effectiveness of the reported method. Obtained results confirm the convergence and spectral accuracy of the proposed method in both space and time. In addition, a comparison with the existing studies is also made as a limiting case of the considered problem at the end and found in good agreement.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 19–29, 2018     

Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models

In this paper nonstandard finite difference (NSFD) schemes of two metapopulation models are constructed. The stability properties of the discrete models are investigated by the use of the Lyapunov stability theorem. As a result of this we have proved that the NSFD schemes preserve essential properties of the metapopulation models (positivity, boundedness and monotone convergence of the solutions, equilibria and their stability properties). Especially, the basic reproduction number of the continuous models is also preserved. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes. The method of Lyapunov functions proves to be much simpler than the standard method for studying stability of the discrete metapopulation model in our very recent paper.     

Impact of the fourth‐order compact difference scheme on computational properties of 3D grids in a nonhydrostatic model

To investigate the potential of the fourth‐order compact difference scheme within the specific context of numerical atmospheric models, a linear baroclinic adjustment system is discretized, using a variety of candidates for practically meaningful staggered 3D grids. A unified method is introduced to derive the dispersion relationship of the baroclinic geostrophic adjustment process. Eight popular 3D grids are obtained by combining contemporary horizontal staggered grids, such as the Arakawa C and Eliassen grids, with optimal vertical grids, such as the Lorenz and Charney‐Phillip (CP) grids, and their time‐staggered versions. The errors produced on the 3D grids in describing the baroclinic geostrophic adjustment process relative to the differential case are compared in terms of frequency and group velocity components with the elimination of implementation error. The results show that by utilizing the fourth‐order compact difference scheme with high precision, instead of the conventional second‐order centered difference scheme, the errors in describing baroclinic geostrophic adjustment process decrease but only when using the combinations of the horizontally staggered Arakawa C grid and the vertically staggered CP or the C/CP grid, the time‐horizontally staggered Eliassen (EL) grid and vertically staggered CP or the EL/CP grid, the C grid and the vertically time‐staggered versions of Lorenz (LTS) grid or C/LTS grid, EL grid and LTS grid or EL/LTS grid. The errors were found to increase for specific waves on the rest grids. It can be concluded that errors produced on the chosen 3D grids do not universally decrease when using the fourth‐order compact difference scheme; hence, care should be taken when implementing the fourth‐order compact difference scheme, otherwise, the expected benefits may be offset by increased errors. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time‐dependent Navier–Stokes equations

This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two‐dimensional non‐stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.     

关于三个非扩张映射收敛于其公共不动点的具误差的三重迭代法

在一致凸Banach空间中,研究了关于三个不同非扩张映射的具误差的三步迭代算法,并在一定条件下证明了此迭代序列收敛于其公共不动点的弱强收敛定理.     

Improvement of the WENO scheme smoothness estimator

This paper has analyzed the weights of the fifth‐order weighted essentially nonoscillatory scheme suggested by Jiang and Shu and a modified smoothness estimator is suggested to improve the accuracy. Using a function of the minimum and maximum of the original smoothness estimators, the new smoothness estimators have smaller variation among them than the original ones in smooth regions. Hence, as the new weights are closer to the optimal weights, the numerical accuracy is improved. Several numerical experiments are presented to demonstrate the accuracy and robustness of the new scheme. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems

In this paper, a new type of anticipating synchronization, called time-varying anticipating synchronization, is defined firstly. Then novel adaptive schemes for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems are designed based on the Lyapunov function and invariance principle. The update gain of coupling strength can be automatically adapted to a suitable strength depending on the initial values and can be properly chosen to adjust the speed of achieving synchronization, so these schemes are analytical and simple to implement in practice. A classical chaotic dynamical system is used to demonstrate the effectiveness of the proposed adaptive schemes with or without parameter uncertainties.     

半离散Jacobi-球面调和谱格式用于流体低马赫数流动

本文提出了一种用于解决流体在球内以低马赫数流动问题的半离散Jacobi-球面调和谱格式,并证明了它的稳定性和收敛性.所用技巧也可应用于球形区域上其它非线性问题.