广义sine-Gordon方程高精度隐式紧致差分方法

本文研究一类二维非线性的广义sine-Gordon(简称SG)方程的有限差分格式.首先构造三层时间的紧致交替方向隐式差分格式,并用能量分析法证明格式具有二阶时间精度和四阶空间精度.然后应用改进的Richardson外推算法将时间精度提高到四阶.最后,数值算例证实改进后的算法在空间和时间上均达到四阶精度.     

Size-dependent resonance and buckling behavior of nanoplates with high-order surface stress effects

This work presents a theoretical study of the resonance frequency and buckling load of nanoplates with high-order surface stress model. A classical thin plate theory based on Kirchhoff–Love assumption is implemented with surface effects. Circular and rectangular nanoplates with simply supported end conditions are exemplified. The size-dependent solutions are compared with the simplified solutions based on simple surface stress model, and also on the classical theory of elasticity. We aim to explore the scope of applicability so that the modified continuum mechanics model could serve as a refined approach in the prediction of mechanical behavior of nanoplates.     

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

     

Secret Sharing Schemes with Detection of Cheaters for a General Access Structure

In a secret sharing scheme, some participants can lie about the value of their shares when reconstructing the secret in order to obtain some illicit benefit. We present in this paper two methods to modify any linear secret sharing scheme in order to obtain schemes that are unconditionally secure against that kind of attack. The schemes obtained by the first method are robust, that is, cheaters are detected with high probability even if they know the value of the secret. The second method provides secure schemes, in which cheaters that do not know the secret are detected with high probability. When applied to ideal linear secret sharing schemes, our methods provide robust and secure schemes whose relation between the probability of cheating and the information rate is almost optimal. Besides, those methods make it possible to construct robust and secure schemes for any access structure.     

Optimal Runge–Kutta smoothers for the p-multigrid discontinuous Galerkin solution of the 1D Euler equations

This work presents a family of original Runge–Kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid solution algorithm. The design criterion for the construction of the Runge–Kutta methods here developed is different form the one traditionally used to derive optimal Runge–Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented.     

Third order residual distribution schemes for the Navier–Stokes equations

We construct a third order multidimensional upwind residual distribution scheme for the system of the Navier–Stokes equations. The underlying approximation is obtained using standard P² Lagrange finite elements. To discretise the inviscid component of the equations, each element is divided in sub-elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub-element. The residuals are distributed to the nodes of each sub-element in a multi-dimensional upwind way. To obtain a discretisation of the viscous terms consistent with this multi-dimensional upwind approach, we make use of a Petrov–Galerkin analogy. The analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms. The performance of this high-order method is tested on flows with high and low Reynolds number.     

双曲型守恒律的一种三阶半离散中心迎风格式

对一维双曲型守恒律,给出了一种具有较小数值耗散的三阶半离散中心迎风格式.该格式以Liu和Tadmor提出的三阶无振荡重构为基础,同时考虑了波传播的单侧局部速度.时间离散用保持强稳定性的三阶Runge-Kutta方法.由于不需用Riemann解算器,避免了特征分解过程,保持了中心格式简单的优点.数值算例验证本方法可进一步减小数值耗散,提高分辨率.     

Remarks on the asymptotic behavior of scalar auxiliary variable (SAV) schemes for gradient-like flows

We introduce a time semi-discretization of a damped wave equation by a SAV scheme with second order accuracy. The energy dissipation law is shown to hold without any restriction on the time step. We prove that any sequence generated by the scheme converges to a steady state (up to a subsequence). We notice that the steady state equation associated to the SAV scheme is a modified version of the steady state equation associated to the damped wave equation. We show that a similar result holds for a SAV fully discrete version of the Cahn-Hilliard equation and we compare numerically the two steady state equations.