第一类边界对流扩散方程LDG方法的稳定性

针对一维常系数对流扩散模型方程,讨论了当含有第一类边界条件时,局部间断有限元方法（LDG方法）的稳定性。利用有限元理论基本分析技巧,证明了当边界条件为第一类的边界条件时,LDG方法为稳定的,并利用数值算例证明理论分析的正确性。     

A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted $L^2$- and $L^{\infty }$-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order $min\{2?\alpha ,1+\alpha \}$, where $\alpha \in (0,1)$ is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.     

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems

In this paper a dissipative exponentially-fitted method for the numerical integration of the Schrödinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrödinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrödinger equation indicates the power of the new approach.     

A free boundary approach and keller's box scheme for BVPs on infinite intervals

A free boundary approach for the numerical solution of boundary value problems (BVPs) governed by a third-order differential equation and defined on infinite intervals was proposed recently [SIAM J. Numer. Anal., 33 (1996), pp. 1473–1483]. In that approach, the free boundary (that can be considered as the truncated boundary) is unknown and has to be found as part of the solution. This eliminates the uncertainty related to the choice of the truncated boundary in the classical treatment of BVPs defined on infinite intervals. In this article, we investigate some open questions related to the free boundary approach. We recall the extension of that approach to problems governed by a system of first-order differential equations, and for the solution of the related free boundary problem we consider now the reliable Keller's box difference scheme. Moreover, by solving a challenging test problem of interest in foundation engineering, we verify that the proposed approach is applicable to problems where none of the solution components is a monotone function.     

A survey on various computational techniques for nonlinear elliptic boundary value problems

In this paper a classification and a survey on numerical techniques for solving nonlinear (quasilinear, semilinear, superlinear, sublinear) elliptic boundary value problems between 2001 and 2006 have been presented and discussed the nature of positive solution of the various problems. The introduction of the methods and results presented by different researchers are summarized.     

A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients

We propose a compact split-step finite difference method to solve the nonlinear Schrödinger equations with constant and variable coefficients. This method improves the accuracy of split-step finite difference method by introducing a compact scheme for discretization of space variable while this improvement does not reduce the stability range and does not increase the computational cost. This method also preserves some conservation laws. Numerical tests are presented to confirm the theoretical results for the new numerical method by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and Gross-Pitaevskii equation.