A NOTE ON STABILITY OF THE SPLIT-STEP BACKWARD EULER METHOD FOR LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS

In the literature （Tan and Wang, 2010）, Tan and Wang investigated the convergence of the split-step backward Euler （SSBE） method for linear stochastic delay integro-differential equations （SDIDEs） and proved the mean-square stability of SSBE method under some condition. Unfortu- nately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.

EXACT AND DISCRETIZED DISSIPATIVITY OF THE PANTOGRAPH EQUATION

The analytic and discretized dissipativity of nonlinear infinite-delay systems of the form x＇（t） = g（x（t）,x（qt））（q∈ （0, 1）, t 〉 0） is investigated. A sufficient condition is presented to ensure that the above nonlinear system is dissipative. It is proved the backward Euler method inherits the dissipativity of the underlying system. Numerical examples are given to confirm the theoretical results.

The time discretization in classes of integro-differential equations with completely monotonic kernels:Weighted asymptotic stability

We study discretization in classes of integro-differential equationswhere the functions aj（t）, 1 ≤ j ≤n, are completely monotonic on （0, ∞） and locally integrable, but not constant. The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term. The stability properties of the discretization are derived in the weighted 11 （p; 0, ∞） norm, where p is a given weight function. Applications to the weighted l^1 stability of the numerical solutions of a related equation in Hilbert space are given.

Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrodinger-Poisson System

We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.

ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR FULLY DISCRETE SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems.Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto[25],a residual based a posteriori error estimators for the state,co-state and control variables are derived.The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements,whereas the piecewise constant functions are employed for the control variable.The temporal discretization is based on the backward Euler method.We derive a posteriori error estimates for the state,co-state and control variables in the L^(∞)(0,T;L^(2)(Ω))-norm.Finally,a numerical experiment is performed to illustrate the performance of the derived estimators.

A MODIFIED WEAK GALERKIN FINITE ELEMENTMETHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION-REACTION PROBLEMS

In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,respectively.We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh.The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms.We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method.Various numerical examples are presented to support the theoretical results.It is theoretically and numerically shown that the method is quite stable.

A SIMPLE DISCRETE-TIME ANALOGUE PRESERVING THE GLOBAL STABILITY OF A CONTINUOUS SIRS EPIDEMIC MODEL

In this paper, we consider the backward Euler discretization derived from a continuous SIRS epidemic model, which contains a remaining problem that our discrete model has two solutions for infected population; one is positive and the other is negative. Under an additional positiveness condition on infected population, we show that the backward Euler discretization is one of simple discrete-time analogue which preserves the global asymptotic stability of equilibria of the corresponding continuous model.