Singularly perturbed boundary value

Alekseevskii, M. (1984) Difference schemes of higher-order accuracy for some singular-perturbed boundary-value problems. Differential Equations, 17, 1177-1183 (in Russian). 

P. A. Markowich and C. A. Ringhofer, A singularly perturbed boundary value problem modelling a semiconductor device, SIAM J. Appl. Math., 44 (1984), pp. 231-256. 

NUMERICAL METHODS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS MODELING DIFFUSION PROCESSES... 

In Section I we obtained an intuitive impression of the numerical problems appearing when one uses classical finite difference schemes to solve singularly perturbed boundary value problems for ordinary differential equations. In this section, for a parabolic equation, we study the nature of the errors in the approximate solution and the normalized diffusion flux for a classical finite difference scheme on a uniform grid and also on a grid with an arbitrary distribution of nodes in space. We find distributions of the grid nodes for which the solution of the finite difference scheme approximates the exact one uniformly with respect to the parameter. The efficiency of the new scheme for finding the approximate solution will be demonstrated with numerical examples. 

We consider the simplest meaningful example leading to singularly perturbed boundary value problems. Suppose that we want to find the distributions of concentration C of a substance in a homogeneous material or in a solid material layer with thickness L. Suppose that the quantity C depends only on the variable y, which is the distance to one side of the material, and that generally speaking, the quantity C varies in time T, that is, C — C y,r). Assume also that inside the material the distributed sources of the substance have a density F y, t). Suppose that the diffusion coefficient D is constant. In this case, the distribution of the substance in a material layer is described by the diffusion equation... 

Thus, by numerical experiments we verify that the approximate solution of the Dirichlet problem (2.16), found by the classical finite difference scheme (2.28), (2.27), and the computed normalized diffusion fiux converge for N, Nq respectively, to the solution of the boundary value problem and the real normalized diffusion flux for fixed e. However, we can also see that they do not converge e-uniformly. The solution of the grid problem approaches the solution of the boundary value problem uniformly in e qualitatively well. The normalized flux computed according to the solution of the difference problem does not approach e-uniformly the real normalized flux (i.e., the flux related to the solution of the boundary value problem) even qualitatively. Nevertheless, if the solution of the singularly perturbed boundary value problem is smooth and e-uniformly bounded, the approximate solution and the computed normalized flux converge e-uniformly (when N, Nq oo) to the exact solution and flux. 

We begin our consideration with a simple example that brings us to the singularly perturbed boundary value problems considered in this section. Suppose that it is required to find the function C(y, t), which is the distribution of temperature in a homogeneous material, or in a layer of solid material with a thickness 2L. As in Section II.A, we assume that the distributed heat sources of density F y, t) act inside the material. Besides these sources, in the middle part of the material a concentrated source of the strength Q(t) is situated at y = 0. In a simplified variant, the temperature distribution in a layer of material is described by the heat equation... 

Many processes of heat and mass transfer, for example, fast-running processes, lead to the investigation of singularly perturbed boundary value problems with a perturbation parameter e. For example, those problems arise in the analysis of heat and mass transfer for mechanical working of materials, in particular, metals. The use of classical methods for the numerical solution of such problems (see, e.g., [1,11,12]) leads us... 

The construction and investigation of special difference schemes for a particular singularly perturbed boundary value problem with a discontinuous initial condition were examined in [16-19]. 

The analysis of heat exchange processes, in the case of the plastic shear of a material, leads us to singularly perturbed boundary value problems with a concentrated source. Problems such as these were considered in Section IV, where it was shown that classical difference schemes give rise to errors, which exceed the exact solution by many orders of magnitude if the perturbation parameter is sufficiently small. Besides, a special finite difference scheme, which allows us to approximate both the solution and... 

Problem (5.13) is a singularly perturbed boundary value problem for a parabolic equation with discontinuous coefficients at the highest order derivative and with a discontinuous initial condition. 

Problem (5.35a) is a singularly perturbed boundary value problem with complicated conditions of exchange on the boundaries of the subdomains. We want to find the solution of the boundary value problem (5.35a) and also the quantity... 

The above examples permit us to reach the following conclusion. For the singularly perturbed boundary value problems arising in the numerical analysis of heat transfer for various technologies, we have constructed special e-uniformly convergent finite difference schemes. These schemes allow us to compute heat fluxes and the quantity of heat transferred across the interfaces of bodies in contact during the processes. Numerical experiments show the efficiency of the new schemes in comparison with classical schemes. 

In the case of singularly perturbed boundary value problems, for which it is required to find diffusion fluxes, computational difficulties arise. These lead to theoretical and applied problems that require special numerical methods allowing us to approximate both the problem solution and the... 

The special finite difference schemes constructed here allow one to approximate solutions of boundary value problems and also normalized di sion fluxes. They can be used to solve effectively applied problems with boundary and interior layers, in particular, equations with discontinuous coefficients and concentrated factors (heat capacity, sources, and so on). Methods for the construction of the special schemes developed here can be used to construct and investigate special schemes for more general singularly perturbed boundary value problems (see, e.g., [4, 17, 18, 24, 35-39]). 

P.A. Farrell, P. W. Hemker, and G. 1. Shishkin, Discrete approximation for a singularly perturbed boundary value problem with parabolic layers, Stichting Mathematisch Centmm, Amsterdam, Report NM-R9502, February 1995. 

G. I. Shishkin, Finite difference approximations for singularly perturbed boundary value problems with diffusion layers. Dublin, Ireland, INCA Preprint No. 2, 1994. 

Numerical Methods for Singularly Perturbed Boundary Value Problems Modeling Diffusion Processes... 

It is known that, in the case of singularly perturbed elliptic equations for which (as the parameter s equals zero) the equation does not contain any derivatives with respect to the space variable, the principal term in the singular part of the solution is described by an ordinary differential equation similar to Eq. (1.16a) (see, e.g., [3-6]). Thus, it can be expected that, when solving singularly perturbed elliptic and parabolic equations using classical difference schemes, one faces computational problems similar to the computational problems for the boundary value problem (1.16). 

In this section, using an example of a boundary value problem for a singularly perturbed ordinary differential equation, we discuss some principles for constructing special finite difference schemes. In Section II.D, these principles will be applied to the construction of special schemes for singularly perturbed equations of the parabolic type. 

In Section II (see Sections II.B, II.D) we considered finite difference schemes in the case when the unknown function takes given values on the boundary. The boundary value problem for the singularly perturbed parabolic equation on a rectangle, that is, a two-dimensional problem, is described by Eqs. (2.12), while the boundary value problem on a segment, that is, a one-dimensional problem, is described by equations (2.14). In Section II.B classical finite difference schemes were analyzed. It was shown that the error in the approximate solution, as a function of the perturbation parameter, is comparable to the required solution for any fine grid. For the above mentioned problems special finite difference schemes were constructed. The error in the approximate solution obtained by the new scheme does not depend on the parameter value and tends to zero as the number of grid nodes increases. 

In this section, we consider singularly perturbed diffusion equations when the diffusion flux is given on the domain boundary. We show (see Section III.B) that the error in the approximate solution obtained by a classical finite difference scheme, depending on the parameter value, can be many times greater than the magnitude of the exact solution. For the boundary value problems under study we construct special finite difference schemes (see Sections III.C and III.D), which allow us to find the solution and diffusion flux. The errors in the approximate solution for these schemes and the computed diffusion flux are independent of the parameter value and depend only on the number of nodes in the grid. 

In Section II.B we saw that for any small step size of the grid a value of the parameter s could be found such that the error in the approximate solution became comparable to the exact solution. In principle, we surmise that eomputational problems can arise also in the case of singularly perturbed equations, when the flux is given on the domain boundary. 

Thus, we see that the newly constructed finite difference schemes are indeed effective and that they allow us to approximate the solution and the normalized diffusion fluxes g-uniformly for both Dirichlet and Neumann boundary value problems with singular perturbations. 

In this section, we consider the singularly perturbed diffusion equation when linear combinations of the solution and its diffusion flux are given on the domain boundary. Such boundary conditions make it possible to realize any of the boundary conditions considered in Sections II and III. Moreover, concentrated sources act inside the domain. These sources lead to the appearance of interior layers. Thus, in addition to the computational problems accompanying the solution of the boundary value problems in Sections II and III, there arise new problems due to the presence of these interior layers. 

On the set G we consider the boundary value problem for the singularly perturbed equation of parabolic type with a concentrated source on 5 ... 

Thus, problem (5.3) is a boundary value problem for a singularly perturbed equation with a concentrated source, which was considered in Section IV. 

In [157] the authors present an initial-value methodology for the numerical approximation of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations. These problems have a boundary layer at one end (left or right) point. The techniaque which used by the authors is to reduce the original problem to an asymptotically equivalent first order initial-value problem. This is done with the... 

M. K. Kadalbajoo and D. Kumar, Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme. Computers <6 Mathematics with Applications, 2009, 57(7), 1147-1156. 

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