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This paper is concerned with the numerical simulations for the dynamics of the time-fractional molecular beam epitaxy models. The variable-step Crank–Nicolson-type schemes are proposed and analyzed for model with and without slope selection respectively. By using the discrete gradient structures of discrete fractional derivative, the numerical schemes preserve discrete variational energy dissipation law on general meshes unconditionally. These are generalizations of classical energy dissipation laws and are asymptotically compatible with the classical MBE model. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our schemes. [ABSTRACT FROM AUTHOR]

In this paper, a center Box difference method is considered for the radially symmetric solution of fractional subdiffusion equation. By method of order reduction, the derivative boundary condition is transformed into Dirichlet boundary condition and thus the geometrical singularity is successfully removed from the original problem. As a matter of course, a natural discretization scheme is obtained. To investigate the stability and convergence of the method, we define a new norm with a weight r d - 1 . Thus, the usual Sobolev inequality is not suitable to the new norm. Therefore, we prove three new Sobolev-like embedding inequalities which can also be applied to the other problems in polar coordinates. Then, the scheme is proved to be unconditionally stable and convergent in maximum norm with the help of the new Sobolev-like embedding inequalities. Some illustrative examples are provided to demonstrate the theoretical results. By some comparisons, it can be seen that the natural discretization scheme is accurate and effective in physical simulations. And it can be used to both long time and short time computation. [ABSTRACT FROM AUTHOR]

Abstract: A linearly localized difference scheme with the first-order time approximation, is proposed for solving a nonlinear Maxwell model associated with the penetration of a magnetic field into a substance. The new scheme is computationally efficient since the resulting algebra equations are linear and can be computed by the fast Thomas algorithm without any Newton-type inner iterations. It is also local in time, that is, only numerical solutions in one previous time-level are necessary to update the current solutions, such that it requires much less storage compared with the fully implicit method. Furthermore, the exponential decaying behavior of difference solution, which is analogous to that of the continuous solution, is obtained. To improve the time accuracy, we apply the Crank–Nicolson-type time discretization to construct a second-order linearly localized method. Numerical examples are presented to support our theoretical results. [Copyright &y& Elsevier]

Abstract: The three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]