The difference methods for the solution of singular-perturbation for the elliptic-parabolic differential equation

In this paper it is discussed the difference method for the solution of singular perturbation problems for the elliptic equations, involving small parameter in the higher derivatives. As ε= 0 the original equations are degenerated into the parabolic equations.Authors constructed special difference schemes by means of the boundary layer properties of the solutions of these problems as well as investigated the convergence of this scheme and asymptotic behaviour of the solutions. Finally, a numerical example is given.     

椭圆-抛物偏微分方程奇异摄动问题的差分解法

本文讨论了含有小参数在高阶导数项的椭圆型方程奇异摄动问题的差分解法.当ε=0时椭圆型方程退化为抛物型方程.作者根据此问题解的边界层性质,构造了特殊的差分格式:研究了它的收敛性和解的渐近性态.最后给出一个数值例题.     

The Difference Methods for the Solution of Singular-Perturbation for the Elliptic-Parabolic Differential Equation

In this paper it is discussed the difference method for the solution of singular perturbation pro-blems for the elliptic equations,involving small parameter in the higher derivatives.Asε=0 theoriginal equations are degenerated into the parabolic equations.Authors constructed special difference schemes by means of the boundary layer properties ofthe solutions of these problems as well as investigated the convergence of this scheme and asymptoticbehaviour of the solutions.Finally,a numerical example is given.     

The boundary layer scheme for a singularly perturbed problem for the second order elliptic equation in the rectangle

Using singularly perturbation theory is constructed the boundary layer scheme for a Dirichlet problem for the second order singularly perturbed equation of elliptic type in the rectangle. The error estimate is given.     

Numerical solution of the singularly perturbed problem for the hyperbolic equation with initial jump

In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.     

Numerical solution of singular perturbation problems for the fourth-order elliptic differential equations

In this paper we construct the finite-difference scheme for the singularly perturbedboundary value problem for the fourth-order elliptic differential equation on the basis ofpaper[1],and prove the uniform convergence of this scheme with respect to the smallparameterεin the discrete energy norm,Finally,we give a numerical example.     

An exponentially fitted difference scheme for the hyperbolic-hyperbolic singularly perturbed initial-boundary value problem

In this paper we discuss,an initial-boundary value problem of hyperbolic type with firstderivative with respect to x.The asymptotic solution is constructed and its uniform validityis proved under weader compatibility conditions.Then we develop an exponentially fitteddifference scheme and establish discrete energy inequality.Finally,we prove that thesolution of difference problem uniformly converges to the solution of the original problem.     

The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L~2 norm is proved and numerical examples are presented.