一类奇摄动拟线性边值问题的激波层现象

研究一类具有内激波层现象的奇摄动拟线性边值问题.在适当的条件下,用合成展开法构造出该问题的一阶形式近似式,并应用不动点原理证明了解的存在性及其当ε→0时的渐近性质.     

A priori meshes for singularly perturbed quasilinear two-point boundary value problems

A quasilinear singularly perturbed boundary value problem withoutturning points is used as a model problem to analyse and comparethe Bakhvalov and Shishkin discretization meshes. The Shishkinmeshes are generalized and improved. Received 26 October 1998. Accepted 3 June 1999.     

Parameter-uniform numerical methods for some singularly perturbed nonlinear initial value problems

Parameter-uniform numerical methods for singularly perturbed nonlinear scalar initial value problems are both constructed and analysed in this paper. The conditions on the initial condition for a stable initial layer to form are identified. The character of a stable initial layer in the vicinity of a double root of the reduced algebraic problem is different to the standard layer structures appearing in the neighbourhood of a single stable root of the reduced problem. Results for a problem where two reduced solutions intersect are also discussed. Numerical results are presented to illustrate the theoretical results obtained.     

On systems of two singularly perturbed quasilinear second-order equations

A system of two quasilinear second-order equations with a small parameter next to the second derivatives is studied. The cases where the matrix of coefficients next to the first derivatives has the following eigenvalues are considered: (a) both of them have negative real parts; (b) they are of opposite sign; (c) one of them is equal to zero. To find the solution and its asymptotics, the initial-value or boundary-value problems are posed depending on the form of these eigenvalues. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 21–28, 2006.     

A brief survey on numerical methods for solving singularly perturbed problems

In the present paper, a brief survey on computational techniques for the different classes of singularly perturbed problems is given. This survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000-2009. However some older important relevant papers are also included in this survey. We also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223-259, 130 (2002) 457-510, 134 (2003) 371-429] for details. Thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems.     

A numerical algorithm for some singularly perturbed boundary value problems

A numerical algorithm is proposed to solve singularly perturbed linear two-point value problems. The method starts with a partial decoupling of the system to obtain two independent subsystems, fast and slow components. Each subsystem is then solved separately. A second-order finite difference scheme is used for this purpose. Numerical examples will be presented to show the efficiency of the method.     

Stability and accuracy of adapted finite element methods for singularly perturbed problems

The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that where u _N is the SDFEM approximation in linear finite element space V ^N of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered. This work was supported in part by NSF DMS-0209497 and NSF DMS-0215392 and the Changjiang Professorship through Peking University.     

Parameter-uniform numerical methods for some linear and nonlinear singularly perturbed convection diffusion boundary turning point problems

Both linear and nonlinear singularly perturbed two point boundary value problems are examined in this paper. In both cases, the problems have a boundary turning point and are of convection-diffusion type. Parameter-uniform numerical methods composed of monotone finite difference operators and piecewise-uniform Shishkin meshes, are constructed and analyzed for both the linear and the nonlinear class of problems. Numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

On the controllability of some singularly perturbed nonlinear systems

The controllability of a large scale dynamic system which depends singularly upon a small parameter λ is considered. When λ = 0, the large scale system degenerates into a reduced order subsystem representing its slow dynamics while neglecting the fast phenomena. Another subsystem, often called a boundary layer system, represents the fast dynamics. In this paper sufficient conditions are established under which the controllability of the overall large scale system is inferred from the same property of the two subsystems.     

Spike-layered solutions of singularly perturbed quasilinear Dirichlet problems

In this paper we study the existence and structure of a least-energy solution for a class of singularly perturbed quasilinear Dirichlet problems. Using the moving plane method we show that this least-energy solution develops to a spike-layer solution on convex domains.     

Spectral methods for some singularly perturbed third order ordinary differential equations

Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations. To solve non-linear equations, Newton’s method of quasi-linearization is applied. The problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other methods available in the literature.      

Layer and unlayer solutions for some singularly perturbed semilinear elliptic systems

In this paper the existence of a transition layer and unlayer solution for the stationary system ɛ ²Δu = f(u, υ), Δυ = g(u, υ) with x ∈ Ω ⊂ R ^N (N ≥ 2) is studied by using the degree-theoretical argument.     

Parameter-uniform numerical methods for singularly perturbed mixed boundary value problems using grid equidistribution

In this paper, we present the analysis of an upwind scheme for obtaining the solution of a convection-diffusion two-point boundary value problem with Robin boundary conditions. The solution is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme converges uniformly with respect to the perturbation parameter. Numerical results are presented that demonstrate the sharpness of the theoretical estimates.     

Analysis of emitter-coupled multivibrators by singularly perturbed systems

Emitter-coupled multivibrators play a decisive role in electrical engineering, especially for phase locked loops which are key-building blocks of analogue RF front-ends. Since multivibrators correspond to relaxation oscillators, in the following the modelling and analysis by the theory of singularly perturbed systems is presented. Models for fast and slow phenomena are derived, and the fast transients of emitter-coupled multivibrators are analysed for the first time. The results of our analysis lead to significant advantages for the design of electrical multivibrators.     

Solving a partially singularly perturbed initial value problem on Shishkin meshes

A coupled first order system of one singularly perturbed and one non-perturbed ordinary differential equation with prescribed initial conditions is considered. A Shishkin piecewise uniform mesh is constructed and used, in conjunction with a classical finite difference operator, to form a new numerical method for solving this problem. It is proved that the numerical approximations generated by this method are essentially first order convergent in the maximum norm at all points of the domain, uniformly with respect to the singular perturbation parameter. Numerical results are presented in support of the theory.     

Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems

Niall Madden ^{We consider a system of 2 one-dimensional singularly perturbedreaction–diffusion equations coupled at the zero-orderterm. The second derivative of each equation is multiplied bya distinct small parameter. We show how to decompose the solutionto the problem into regular and layer parts. Properties of thediscretized operator are established using discrete Green'sfunctions. We prove that a central difference scheme on certainlayer-adapted meshes converges independently of the perturbationparameters. Supporting numerical examples confirm our theoreticalresults.}     

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

The objective of this paper is to construct and analyze a fitted operator finite difference method (FOFDM) for the family of time‐dependent singularly perturbed parabolic convection–diffusion problems. The solution to the problems we consider exhibits an interior layer due to the presence of a turning point. We first establish sharp bounds on the solution and its derivatives. Then, we discretize the time variable using the classical Euler method. This results in a system of singularly perturbed interior layer two‐point boundary value problems. We propose a FOFDM to solve the system above. Through a rigorous error analysis, we show that the scheme is uniformly convergent of order one with respect to both time and space variables. Moreover, we apply Richardson extrapolation to enhance the accuracy and the order of convergence of the proposed scheme. Numerical investigations are carried out to demonstrate the efficacy and robustness of the scheme.